MASARYK UNIVERSITY
FACULTY OF SCIENCE
QUALITATIVE THEORY
OF FRACTIONAL DIFFERENTIAL
SYSTEMS WITH TIME DELAY
HABILITATION THESIS
Ing. TOMÁŠ KISELA, Ph.D.
BRNO 2024
ii
copyright © 2024, Tomáš Kisela
Preface
The idea behind derivatives of non-integer orders traces back nearly as far as the
classical derivatives themselves. What began in the 17th century as a puzzling
thought exercise has evolved into one of the most influential mathematical fields of
recent decades, known as fractional calculus. My background in both mathematics
and physics continues to feed my interest in this elegant discipline and its impact
on both theory and applications.
In the realm of applied sciences, fractional calculus has gained recognition for its
linear descriptions of complex systems characterized by nonlocal or memory-based
behaviour, traditionally treated within the nonlinear domain. Theoretically, the
ability to continuously transition between derivative orders reveals previously unseen
connections and enables the study of various phenomena. However, the generalizing
nature of fractional calculus also poses a risk: researchers are tempted to deal with
artificial, easily solvable problems that neither enrich applications nor contribute
much to theory, resulting in mere formalism. With this in mind, I have made it
my goal to focus on key problems that help to shed light on deeper mathematical
principles and behaviour of complex systems.
This habilitation thesis combines the key results of my seven selected papers
[6, 10–13, 15, 37] from 2016-2023. Their unifying theme is the qualitative analysis
of fractional delay differential equations, a class of mathematical models involving
fractional derivatives and time delays representing inherent lags in the system. The
key aim is two-fold: first, to better understand, predict and control the behaviour of
such systems, which is essential for numerous applications ranging from physics and
biology to engineering and finance. Second, to integrate fractional calculus into the
broader landscape of dynamic systems theory, where it can illuminate the intricate
interplay of delayed responses and memory effects in shaping system behaviour.
The thesis comprises four chapters that present the main results and provide
commentary on key proofs referring the respective papers, and seven appendices
containing the full text of the selected published papers. Chapter 1 provides a
context of classical and fractional qualitative analysis and introduces known limit
cases of our study. Chapter 2 summarizes the original results on one-term fractional
delay differential systems. It sets the theoretical foundation for subsequent work
and presents key stability and oscillatory conditions, often in non-improvable form.
Chapter 3 focuses on two-term fractional delay differential equations. It explores
the relationships between derivative order and the stability regions for the system’s
coefficients. Finally, Chapter 4 offers concluding remarks and reflections on this
– iii –
iv
research.
I express deep gratitude to Jan Čermák, my former advisor, leader of the scientific
group I am proud to be part of and my most frequent co-author. His insights,
leadership, continuous support and willingness to share his broad mathematical
knowledge were essential for the completion of this work. I also thank my other coauthors
and colleagues for their collaboration, namely Zuzana Došlá, Jan Horníček
and Luděk Nechvátal who participated in some of the seven papers forming the base
of this thesis. I extend my thanks to Alberto Cabada and Matej Dolník who influenced
my thinking during joint work on other topics. In addition to the professional
support I have received, I am especially grateful to Lída Kiselová, my dear wife, and
to my children Bára, Magda and Teodor for their endless patience, encouragement
and understanding without which I could not have accomplished this work. I thank
my parents, grandparents and entire family as well as my friends for an occasional
gentle push that kept me moving forward.
Brno, September 2024 Tomáš Kisela
Contents
1 Wider context: classical and fractional qualitative analysis 1
1.1 Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Classical results of qualitative analysis . . . . . . . . . . . . . . . . . 3
1.3 Classical characteristic equation approach . . . . . . . . . . . . . . . 7
2 Analysis of one-term fractional delay differential systems 9
2.1 Structure of solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 Asymptotics of generalized delay exponentials . . . . . . . . . . . . . 12
2.3 Decomposition of eigenvalues’ complex plane . . . . . . . . . . . . . . 14
2.4 Asymptotically stable systems . . . . . . . . . . . . . . . . . . . . . . 16
2.5 Unstable systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3 Analysis of two-term fractional delay differential equations 23
3.1 Structure and asymptotics of solutions . . . . . . . . . . . . . . . . . 24
3.2 Stability regions for orders less than one . . . . . . . . . . . . . . . . 26
3.3 Stability regions for orders less than one and an imaginary coefficient 29
3.4 Stability regions for orders from one to two . . . . . . . . . . . . . . . 33
4 Conclusions 39
Bibliography 41
Appendices 45
A Lower-order one-term FDDS (CNSNS, 2016) 47
B Higher-order one-term FDDS (EJDE, 2019) 65
C Overview for one-term FDDS (Math Appl, 2020) 81
D Lambert function and one-term FDDE (FCAA, 2023) 95
E Lower-order two-term FDDE (AMC, 2017) 117
F Lower-order complex two-term FDDE (CNSNS, 2019) 133
G Higher-order two-term FDDE (CNSNS, 2023) 153
– v –
Contents vi
Chapter 1
Wider context: classical and
fractional qualitative analysis
The study of dynamic systems involving time delays is a classical area of mathematical
analysis with significant real-world applications. These systems can accurately
model processes that do not respond instantaneously to changes in their state or environment,
such as biological systems, economic models, and engineering processes.
Despite extensive research efforts, many questions related to the stability and control
of these classical systems remain unanswered, primarily due to the inherent
challenges of incorporating delayed responses (see, e.g. [21,22,38]).
Fractional calculus (extending the concepts of differentiation and integration to
non-integer orders) is known for its ability to address the complexities of dynamic
systems with nonlocal and memory effects, such as systems where the future state
depends on a continuum of past states. The corresponding, so-called fractional,
dynamic systems attract a significant attention of scientific community for several
decades (see, e.g. [25,26,32,48,50,53]). Many qualitative results of classical calculus
already found their fractional counterparts (in particular in linear case), many wait
for further progress and many may be impossible to generalize.
The developing interest in fractional calculus among scientists and engineers is
largely due to its applications and further potential in control theory. Thus, the
need to model the fractional systems with delayed feedback with sufficient precision
is growing (see [20,27,49,52]). In particular, since there appears to be a clash of two
forces: while fractional systems of lower orders typically show larger stability regions,
growing delay tends to destabilize the system. That is why this thesis focuses on
the domain connecting fractional derivatives and time delays, on the qualitative
theory of linear fractional delay differential systems (FDDS). Study of FDDS serves,
besides its theoretical value, as a mathematical basis for effective feedback control
of complex systems with memory.
This chapter sets the stage by providing the necessary context. We recall basic
notions of fractional calculus and qualitative theory, present some classical results
serving as comparisons later in the text and outline the basic ideas behind qualitative
analysis.
– 1 –
Chapter 1. Wider context: classical and fractional qualitative analysis 2
1.1 Basic notions
As mentioned above, the subject of this thesis is a study of systems involving fractional
derivative. Throughout the text we utilize the following definitions: Let a be
a real number and let f be a real scalar function defined on (a, ∞). Its fractional
integral of positive real order γ is given by
D−γ
a f(t) =
t
a
(t − ξ)γ−1
Γ(γ)
f(ξ)dξ, t ∈ (a, ∞) .
The (Caputo) fractional derivative of positive real order α is given by
Dα
a f(t) = D−(⌈α⌉−α)
a
d⌈α⌉
dt⌈α⌉
f(t) , t ∈ (a, ∞) (1.1)
where ⌈·⌉ denotes an upper integer part (so-called ceiling function). As it is customary,
we put D0
af(t) = f(t). Besides the Caputo definition (1.1) employed throughout
this thesis, some authors use the Riemann-Liouville derivative applying the fractional
integral before the integer-order derivative. We mention this approach only
occasionally for comparison. For more basics of fractional calculus we refer to [32,53].
If f is a vector function, the corresponding fractional operators are considered
component-wise, if f is a complex-valued function, the corresponding fractional
operators are introduced for its real and imaginary part separately.
The terminology that we employ is based on the classical qualitative theory:
• A linear differential system is said to be stable if all its solutions are bounded
as t → ∞.
• A linear differential system is said to be asymptotically stable if all its solutions
tend to zero as t → ∞.
• The set of all parameters’ values for which the differential system is asymptotically
stable is called the stability region.
• The solution to a differential system is called oscillatory if its set of zeros is
unbounded.
For more precise large-time solution descriptions we use the following asymptotic
notations (K being a suitable positive real):
f ∼ g as t → ∞ ⇐⇒ lim
t→∞
f(t)
Kg(t)
= 1 ,
f ∼sup g as t → ∞ ⇐⇒ lim sup
t→∞
f(t)
Kg(t)
= 1 ,
f = O(g) as t → ∞ ⇐⇒ lim sup
t→∞
|f(t)|
g(t)
< ∞ ,
f = o(g) as t → ∞ ⇐⇒ lim sup
t→∞
f(t)
g(t)
= 0 .
In addition to the asymptotic equivalence ∼, they enable us to describe asymptotics
of a wider class of functions (in particular, unbounded oscillatory functions).
3 Chapter 1. Wider context: classical and fractional qualitative analysis
1.2 Classical results of qualitative analysis
In this section, we summarize the most important results regarding qualitative properties
of fractional differential systems without delay, ordinary delay differential systems
and ordinary differential systems with both delayed and undelayed terms. In
particular, we recall the inequalities defining stability regions for the corresponding
problems. We also provide commentary to other relevant sources expanding these
results.
Fractional differential systems without delay
Let us consider the system
Dα
0 y(t) = Ay(t) , t ∈ [0, ∞) (1.2)
where A is a constant real d×d matrix and α > 0. The classic stability result comes
from [45] and corresponds to the following assertion.
Theorem 1.1. Let A ∈ Rd×d
, α ∈ R+
\ Z+
and let λj (j = 1, 2, . . . , d) be all
eigenvalues of A. Then (1.2) is asymptotically stable if and only if
0 < α < 2 and |Arg (λj)| > απ/2 for all j .
Moreover, any solution y tends to zero algebraically as y ∼ t−α
as t → ∞.
Corollary 1.2. The stability region of (1.2) is given by
Sα = {λ ∈ C : |Arg (λ)| > απ/2} , 0 < α < 2 .
Remark 1.3. The proof was given in [45] and the used proving technique enables
to show several other assertions:
a) Originally, only the order less than one was considered, however, the technique
works analogously for higher orders.
b) It was also proven that (1.2) is stable if and only if 0 < α < 2 and |Arg (λj)| ≥
απ/2 for all j and those eigenvalues with the principal argument equaling to ±απ/2
have geometric multiplicity one.
c) Unbounded solutions of (1.2) follow the asymptotic relation y ∼sup tk
exp(λ1/α
t)
where k is a suitable nonnegative integer.
d) All solutions of (1.2) tending to zero as t → ∞ are non-oscillatory. For other
solutions, oscillatory property might occur.
Figures 1.1 and 1.2 illustrate the evolution of the stability region Sα for increasing
α. We note that for α → 1 the stability boundary coincides with imaginary axis
and for α → 2−
the stability region degenerates into an empty set. That shows
the agreement with the classical theory for both first and second order differential
systems.
In case of the scalar equation, i.e. Dα
0 x(t) = λx(t) with λ being real, the asymptotic
stability and stability occur for λ < 0 and λ ≤ 0, respectively (provided
α ∈ (0, 2)).
Chapter 1. Wider context: classical and fractional qualitative analysis 4
ℜ
ℑ
2
2
Figure 1.1: Stability region Sα for
(1.2) with α = 0.4 < 1.
ℜ
ℑ
Figure 1.2: Stability region Sα for
(1.2) with α = 1.4 > 1.
First-order delay differential system
Let us consider the system with time delay
y′
(t) = Ay(t − τ), t ∈ [0, ∞), (1.3)
where A is a constant real d × d matrix and τ > 0 is a constant real lag. Its main
stability and asymptotic properties were derived in, e.g. [23] and can be formulated
as
Theorem 1.4. Let A ∈ Rd×d
, τ ∈ R+
and let λj (j = 1, 2, . . . , d) be all eigenvalues
of A. Then (1.3) is asymptotically stable if and only if
τ|λj| < |Arg (λj)| − π/2 for all j .
Moreover, any solution y tends to zero exponentially as t → ∞.
Corollary 1.5. The stability region of (1.3) is given by
Sτ
1 = λ ∈ C : |λ| <
|Arg (λ)| − π/2
τ
, |Arg (λ)| >
π
2
, τ > 0 .
Oscillation properties of (1.3) can be written as follows (see, e.g. [21]).
Theorem 1.6. Let A ∈ Rd×d
, τ ∈ R+
and let λj (j = 1, 2, . . . , d) be all eigenvalues
of A. Then all solutions of (1.3) oscillate if and only if
λj ∈ C \ [−1/(τe), ∞) for all j ,
i.e. A has no real eigenvalues in [−1/(τe), ∞).
5 Chapter 1. Wider context: classical and fractional qualitative analysis
ℜ
ℑ
1
Figure 1.3: Stability region Sτ
1 for
(1.3) depicted for the value τ = 1.
ℜ
ℑ
0
Figure 1.4: Stability region Sτ
0 for
(1.4), independent of τ.
Figure 1.3 shows the stability region for (1.3), highlighting that in the case of
scalar equation x′
(t) = λx(t−τ) with λ being real, the asymptotic stability condition
reduces to −π/(2τ) < λ < 0. Also, in scalar case all solutions oscillate if and only
if λ < −1/(τe).
Discrete system
Let us consider the discrete system
y(n) = Ay(n − τ), t ∈ {τ, 2τ . . . }, (1.4)
where A is a constant real d × d matrix and τ > 0 is a constant real lag. This
system can be viewed as a modification of (1.3) where the derivative is removed (the
derivative order is changed to zero) and the time is discretized into multiples of τ.
It is well-known that the corresponding stability region (see Figure 1.4) is given by
a unit circle with no dependence on the value of τ, i.e.
Sτ
0 = {λ ∈ C : |λ| < 1} , τ > 0 .
First-order differential equation with both delayed and undelayed terms
If an undelayed term is added to the right-hand side of (1.3), we obtain a system
for which the stability analysis is quite difficult even in the planar case. Corresponding
necessary and sufficient stability conditions given in terms of system parameters
are known only in very special cases, e.g. if A, B are simultaneously triangularizable.
For more details see [4,31,46]. Hence, regarding right-hand side composed of both
delayed and undelayed terms, we will focus on scalar equations.
Let us consider the equation
y′
(t) = ay(t) + by(t − τ), t ∈ [0, ∞), (1.5)
Chapter 1. Wider context: classical and fractional qualitative analysis 6
where a, b are real and τ > 0 is a constant real lag. Its stability properties can be
written as (see [24])
Theorem 1.7. Let a, b ∈ R, τ ∈ R+
. Then (1.5) is asymptotically stable if and
only if either
a ≤ b < −a and τ is arbitrary ,
or
|a| + b < 0 and τ <
arccos(−a/b)
(b2 − a2)1/2
.
Corollary 1.8. The stability region of (1.5) is given by
Sτ
1 = (a, b) ∈ R2
: a − b ≤ 0 and a + b < 0
∪ (a, b) ∈ R2
: |a| + b < 0 and τ <
arccos(−a/b)
(b2 − a2)1/2
.
Figure 1.5 displays the stability region for (1.5) in the (a, b)–plane. The top part
of the stability boundary is formed by the axis of the second and fourth quadrants
corresponding to the first condition in Theorem 1.7. The bottom part of the stability
boundary representing the second condition in Theorem 1.7 depends on τ as also
illustrated by the cusp point coordinates.
a
b
1
[ [;
Figure 1.5: Stability region Sτ
1 for
(1.5) depicted for the value τ = 1.
a
b
2
Figure 1.6: Stability region Sτ
2 for
(1.6) depicted for the value τ = 1.
Second-order differential equation with both delayed and undelayed terms
For better comparison, let us also consider the second-order system
y′′
(t) = ay(t) + by(t − τ), t ∈ [0, ∞), (1.6)
7 Chapter 1. Wider context: classical and fractional qualitative analysis
where a, b are real and τ > 0 is a constant real lag. Its stability properties can
be derived from [5] (although the case b < 0 was not explicitly discussed there) to
obtain
Theorem 1.9. Let a, b ∈ R, τ ∈ R+
and ℓ ∈ Z+
0 be such that
ℓ2 π2
τ2
< |a| < (ℓ + 1)2 π2
τ2
.
Then (1.6) is asymptotically stable if and only if a < 0 and either
0 < b < min(−ℓ2 π2
τ2
− a, (ℓ + 1)2 π2
τ2
+ a) for ℓ being zero or even,
or
0 > b > max(ℓ2 π2
τ2
+ a, −(ℓ + 1)2 π2
τ2
− a) for ℓ being odd.
Corollary 1.10. The stability region of (1.6) is given by
Sτ
2 =
∞
j=0
(a, b) ∈ R2
: 0 < b < min(−(2j)2 π2
τ2
− a, (2j + 1)2 π2
τ2
+ a)
∪ (a, b) ∈ R2
: 0 > b > max((2j + 1)2 π2
τ2
+ a, −(2j + 2)2 π2
τ2
− a) .
Comparing Theorems 1.7 and 1.9 we see very different stability conditions for
the first and second-order equations. This difference is demonstrated in Figures 1.5
and 1.6 in the form of quite distinct shapes of the corresponding stability regions.
The transition between them with continuous change of derivative order will be one
of the interest of the following chapters.
1.3 Classical characteristic equation approach
The characteristic equation is central to the stability analysis of linear differential
systems, including those with delays. The usual approach involves substituting
an exponential function with argument st (where s is a complex parameter) as a
candidate solution into the system. This substitution transforms the differential
system into an algebraic equation with s as the variable, such as
det(sI − A exp(−sτ)) = 0 for (1.3) ,
s − a − b exp(−sτ) = 0 for (1.5) ,
s2
− a − b exp(−sτ) = 0 for (1.6) .
Unlike the characteristic equations of ordinary differential equations, which are polynomial
in s, these equations are transcendental, leading to more challenges as they
typically have an infinite number of roots.
The system stability is then determined by the location of the characteristic roots
in the complex plane due to the well-known behaviour of exponential functions:
Chapter 1. Wider context: classical and fractional qualitative analysis 8
• If all roots have negative real parts, the system is asymptotically stable.
• If any root has a positive real part, the system is unstable.
• If the rightmost root, i.e. the root with the largest real part, lies on the imaginary
axis, there might be stability or instability based on root multiplicities.
If entry parameters of the system are specified, the position of characteristic roots
with respect to imaginary axis can be usually analyzed numerically case by case.
However, if we need to design a system or its control, or if there is a risk of parameter
uncertainty, this approach is very random and impractical. Thus, the focal point of
our effort is a reformulation of stability conditions from terms of characteristic roots
into terms of entry parameters.
That is usually done via finding stability boundary in the space of entry parameters.
In other words, we are looking for all combinations of entry parameters
yielding rightmost roots with zero real part. Due to continuous dependence of characteristic
roots on entry parameters, we arrive at a hypersurface in the parameter
space where the system transitions from stable to unstable. This approach is often
called D-partition method, D-decomposition method or boundary locus method
(see, e.g. [24,30,39,46,47,54]).
If we consider a system of non-integer order, there is one significant difference: the
exponential functions do not longer solve the system. We need to find an alternative
way to derive the characteristic equation. The well-established practice is to employ
Laplace transform method (for definition we refer to [17]) which leads to the same
results for all the classical problems and is successfully used for fractional differential
systems as well. In particular, for (1.2) it yields the well-known formula
det(sα
I − A) = 0
illustrating that characteristic equations belonging to fractional differential problems
typically contain non-analytic functions.
Further, we have to ensure the connection between location of characteristic
roots and stability properties of the system other than the exponential argument
(as exponentials are no longer solutions, see, e.g. [46,53]).
As this thesis deals with problems combining both fractional orders and delays,
the main challenges addressed in the following chapters are:
• Investigating the properties of roots of characteristic equations that are transcendental
and non-analytic.
• Identifying efficient descriptions of stability boundaries in various parameter
spaces, clarifying the role of derivative order and delay in shaping the corresponding
stability regions.
• Deriving asymptotic expansions of various special functions, often by using
the inverse Laplace transform.
Chapter 2
Analysis of one-term fractional delay
differential systems
This chapter focuses on the stability, oscillatory and related asymptotic properties
derived in author’s papers [6,10,15,37] for one-term FDDS
Dα
0 y(t) = Ay(t − τ), t ∈ (0, ∞) (2.1)
where A is a constant real d×d matrix and α, τ > 0 are real scalars. The associated
initial conditions have typically the form
y(t) = ϕ(t), t ∈ [−τ, 0] , (2.2)
lim
t→0+
y(j)
(t) = ϕj, j = 0, . . . , ⌈α⌉ − 1 (2.3)
where all components of d-vector function ϕ are absolutely Riemann integrable on
[−τ, 0] and ϕj are constant real d-vectors.
The presence of fractional derivative creates room for discussions regarding the
proper choice of its lower limit a (see (1.1)) which coincides with the "time origin" of
the system. In particular, one might ask why not to put this limit to a = −τ? Similar
issues were discussed as the problem of so-called initialization in [44]. Although this
discussion is quite interesting, no matter the result it does not significantly affect
the qualitative study, because the change of the lower limit is analogous to adding a
forcing term on the right-hand side of (2.1). Hence, we adopt the standard approach
and consider the lower limit of fractional operators to be zero.
The study of qualitative properties of (2.1) was approached by many authors
from different angles. One of the first attempts was [29] dealing with scalar version
of (2.1) with real parameter employing the Lambert function to discuss asymptotic
properties of solutions. Many authors in 2005-2012, e.g. [16,40,54], studied problems
of vector nature, often involving more fractional derivative terms and more time
delays. However, the stability criteria were almost exclusively limited on conditions
for locations of characteristic roots in complex plane without an explicit link to
the entry parameters of the corresponding problem. The difficult practical use and
lack of efficiency of such results were often mentioned by authors themselves (see,
– 9 –
Chapter 2. Analysis of one-term fractional delay differential systems 10
e.g. [16, 40]). To our knowledge, the first explicit stability criterion was published
for scalar version of (2.1) in 2011 by [39].
With this background, we started our work on [6] in 2014 and managed to derive
explicit stability criterion and the asymptotics of bounded solutions for (2.1)
of low orders (less than one). Three years later, in [10], we expanded our scope to
(2.1) of higher orders (greater than one) for which we thoroughly analyzed oscillatory
properties which, to our knowledge, were not discussed to that extent in the
literature at the time (see, e.g. [3]). In 2020, we consolidated these results in [37],
providing a comprehensive summary of the stability and asymptotic properties of
(2.1) across all orders. Additionally, we extended our findings to systems involving
another, so-called Riemann-Liouville, fractional derivative requiring a different type
of initial conditions. It was only in 2023, in [15], when we added an easy-to-use
graphical approach to estimate the asymptotic behaviour of unbounded solutions
based on properties of Lambert function.
The key results from these four papers serve as the foundation for the following
sections. As (2.1) transitions into (1.2) when τ → 0, and reduces to (1.3) as α → 1,
this chapter focuses on comparing the properties of (2.1) with its limit counterparts.
In Section 2.1 we establish the structure of solutions to (2.1) and the role of so-called
generalized delay exponentials whose asymptotic properties are analyzed in Section
2.2. Section 2.3 describes the decomposition of complex plane naturally imposed by
characteristic roots with zero real parts. Finally, Sections 2.4 and 2.5 are devoted
to asymptotically stable and unstable systems, respectively, namely to the evolution
of stability conditions, asymptotics and oscillatory properties with changes in the
derivative order α.
2.1 Structure of solutions
Regarding solving of linear fractional equations, the Laplace transform is one of
the most powerful tools. The problem (2.1)-(2.3) is no different. Applying Laplace
transform (see [6,10,37]), we quickly notice the significance of the following notion:
Definition 2.1. Let A ∈ Rd×d
, let I be the identity d × d matrix and let α, τ ∈ R+
.
The matrix function R : R → Cd×d
given by
R(t) = L−1
(sα
I − A exp{−sτ})−1
(t) (2.4)
is called the fundamental matrix solution of (2.1). We note that L−1
denotes the
standard inverse Laplace transform, i.e. L−1
(F(s))(t) = (2π i)−1 c+i ∞
c−i ∞
F(s)est
ds.
The inverse matrix occurring in (2.4) suggests the well-known characteristic equation
associated to (2.1)
det(sα
I − A exp{−sτ}) = 0 , i.e.
n
i=1
(sα
− λi exp{−sτ})ni
= 0 (2.5)
11 Chapter 2. Analysis of one-term fractional delay differential systems
where λi (i = 1, . . . , n) are distinct eigenvalues of A and ni are their algebraic
multiplicities.
The concept of fundamental matrix solution (for integer orders see, e.g. [22])
yields the following solution representation depending on the initial conditions:
Theorem 2.2. Let A ∈ Rd×d
, α, τ ∈ R+
and R be the fundamental matrix solution
of (2.1). Then the solution y of (2.1)–(2.3) is given by
y(t) =
⌈α⌉−1
j=0
Dα−j−1
0 R(t)ϕj +
0
−τ
R(t − τ − u)Aϕ(u)du .
Proof. The assertion follows directly from the evaluation of inverse Laplace transform
of (2.1)–(2.3), for details see [6,37].
To use these findings for qualitative analysis, we have to find more nuanced
description of the solution. Applying the theory of Jordan canonical matrices on
the fundamental matrix solution, we discover a key role of the function introduced
by
Definition 2.3. Let λ ∈ C, η, β, τ ∈ R+
and m ∈ Z+
∪ {0}. The generalized delay
exponential function (of Mittag-Leffler type) is introduced via
Gλ,τ,m
η,β (t) =
∞
j=0
m + j
j
λj
(t − (m + j)τ)η(m+j)+β−1
Γ(η(m + j) + β)
h(t − (m + j)τ)
where h is the Heaviside step function.
Remark 2.4. We note that special choices of G function parameters yield functions
known to solve special cases of (2.1). Indeed,
• Gλ,0,0
1,1 (t) reduces to classical exponential exp{λt} solving y′
(t) = λy(t),
• Gλ,0,0
α,1 (t) coincides with one-parameter Mittag-Leffler function Eα(λtα
) solving
the scalar version of (1.2) (see, e.g. [53]),
• Gλ,τ,0
1,1 (t) is the delay exponential solving the scalar version of (1.3) (see, e.g. [2]).
The Laplace transform of the generalized delay exponential function of MittagLeffler
type is
L(Gλ,τ,m
η,β (t))(s) =
sη−β
exp{−msτ}
(sη − λ exp{−sτ})m+1
, (2.6)
which allows us to detail the fundamental matrix solution as follows.
Chapter 2. Analysis of one-term fractional delay differential systems 12
Lemma 2.5. The fundamental matrix solution (2.4) can be expressed as R(t) =
T−1
G(t)T, where T is a regular matrix and G is a block diagonal matrix with uppertriangular
blocks Bj given by
Bj(t) =







Gλi,τ,0
α,α (t) Gλi,τ,1
α,α (t) Gλi,τ,2
α,α (t) · · · G
λi,τ,rj−1
α,α (t)
0 Gλi,τ,0
α,α (t) Gλi,τ,1
α,α (t) · · · G
λi,τ,rj−2
α,α (t)
0 0 Gλi,τ,0
α,α (t) · · · G
λi,τ,rj−3
α,α (t)
...
...
...
...
...
0 0 0 · · · Gλi,τ,0
α,α (t)







,
where j = 1, . . . , J (J ∈ Z+
), rj is the size of the corresponding Jordan block of A.
Proof. See [6,10].
Summarizing the above-stated results, we arrive at the crucial assertion describing
the role of G functions, which serves as foundation for our next analysis.
Theorem 2.6. Let R(t) be the fundamental matrix solution of (2.1). Further, let
λi (i = 1, . . . , n) be distinct eigenvalues of A and let pi be the largest dimension of
the Jordan block corresponding to the eigenvalue λi. Then the nonzero elements of
R(t) are given by linear combinations of the generalized delay exponential functions
Gλi,τ,m
α,α (t), m = 0, . . . , pi − 1, i = 1, . . . , n.
2.2 Asymptotics of generalized delay exponentials
As mentioned in Section 1.3, the known asymptotics of exponential functions is
underlying most of the qualitative analysis of integer-order problems. Analogously,
the asymptotic properties of the generalized delay exponential function of MittagLeffler
type (Definition 2.3) prove to be crucial in qualitative analysis of (2.1).
First, let us introduce the real-part ordering for the roots of the denominator in
(2.6) where, for the sake of simplicity, we set η = α (note the link to the characteristic
equation (2.5)). Let sj (j = 1, 2, . . . ) be the roots of
sα
− λ exp{−sτ} = 0
with ordering ℜ(sj) ≥ ℜ(sj+1), particularly s1 is called the rightmost root. Then
we can write the foundational asymptotic result as
Lemma 2.7. Let λ ∈ C, α ∈ R+
\ Z+
, β, τ ∈ R+
, m ∈ Z+
0 and sj be roots of (2.6)
with real-part ordering.
(i) If λ = 0, then
G0,τ,m
α,β (t) =
(t − mτ)mα+β−1
Γ(mα + β)
h(t − mτ).
13 Chapter 2. Analysis of one-term fractional delay differential systems
(ii) If λ ̸= 0, then
Gλ,τ,m
α,β (t) =
∞
j=1
m·kj
ℓ=0
ajℓ(t − mτ)ℓ
exp{sj(t − mτ)} + Pλ,τ,m
α,β (t) ,
where kj is a multiplicity of sj, ajℓ are suitable nonzero complex constants (ℓ =
0, . . . , mkj, j = 1, 2, . . . ) and the term Pλ,τ,m
α,β has the algebraic asymptotic behaviour
expressed via
Pλ,τ,m
α,β (t) =
(−1)m+1
λm+1Γ(β − α)
(t + τ)β−α−1
+
(−1)m+1
(m + 1)
λm+2Γ(β − 2α)
(t + 2τ)β−2α−1
+ O(tβ−3α−1
) as t → ∞.
Proof. For the proof in its complete form see [6], for additional supplementary assertions
needed for higher orders see [10].
Its idea is built around evaluation of G for large t through the inverse Laplace
transform
Gλ,τ,m
α,β (t) =
1
2πi γ(R, π
2
+δ)
sα−β
exp{st − msτ}
(sα − λ exp{−sτ})m+1
ds .
The symbol γ (R, π/2 + δ) denotes the specific oriented piecewise smooth curve (see
Figure 2.1) formed by three segments, i.e. γ(µ, θ) = γ1 + γ2 + γ3 where µ > 0,
θ ∈ (0, π] and
γ1 = {s ∈ C : s = −u exp{−iθ}, u ∈ (−∞, −µ)} ,
γ2 = {s ∈ C : s = ζ exp{−iu}, u ∈ [−π − θ, π + θ]} ,
γ3 = {s ∈ C : s = u exp{−iθ}, u ∈ (µ, ∞)} .
The proof, apart from its considerable technical difficulty, utilizes several properties
of characteristic roots. In particular, there exists δ > 0 such that all roots
sj of (2.5) satisfy | Arg(si)| ̸= π/2 + δ and, moreover, that there are only finitely
many of them satisfying | Arg(si)| < π/2+δ. For detail calculations of relevant root
properties, see [6].
Remark 2.8. Notice that Lemma 2.7 focuses on non-integer values of α. The
cases of integer values are already covered by the classical theory and are known
to have exponential asymptotics. From the technical standpoint, the difference lies
in the fact that for non-integer α the Laplace transform of G contains non-analytic
function, while for integer α only analytic functions occur.
Chapter 2. Analysis of one-term fractional delay differential systems 14
γ
γ
γ
ℜ
ℑ
1
2
3
μ
ϑ
+ϑ
Figure 2.1: The curve γ(µ, θ) used for evaluation of the inverse Laplace
transform in the proof of Lemma 2.7.
2.3 Decomposition of eigenvalues’ complex plane
Lemma 2.7 implies that, similarly to integer-order cases, the characteristic roots
affect the stability properties primarily depending on the sign of their real parts.
Hence, in this section we investigate the relation between locations of system matrix
eigenvalues λ for (2.1) and zero real parts of characteristic roots of (2.5). In particular,
we decompose the complex plane into regions such that eigenvalues chosen
inside these regions guarantee nonzero real parts of the corresponding characteristic
roots, and eigenvalues lying on boundaries of these regions imply at least one
characteristic root with the zero real part.
Applying the standard approach of substituting s = i φ (φ ∈ R) into factors of
(2.5), i.e. sα
− λ exp{−sτ} = 0, equating real and imaginary parts and rearranging
with respect to |λ| and Arg(λ). After a tedious calculations (see [10]), we can
eliminate the parameter φ and define the regions as follows:
For any α > 0 and m ∈ Z+
such that 0 < α < 4m + 2:
Qτ
α(m) = λ ∈ C : |λ| <
| Arg(λ)| − απ
2
+ 2mπ
τ
α
,
απ
2
− 2mπ < | Arg(λ)| ≤
απ
2
− (2m − 2)π
∪ λ ∈ C :
| Arg(λ)| − απ
2
+ (2m − 2)π
τ
α
< |λ| <
| Arg(λ)| − απ
2
+ 2mπ
τ
α
,
| Arg(λ)| >
απ
2
− 2mπ
where the sets Qτ
α(m) (m ∈ Z+
0 ) are defined to be empty whenever α ≥ 4m + 2.
15 Chapter 2. Analysis of one-term fractional delay differential systems
Further, for α ∈ (0, 2) we add:
Qτ
α(0) = λ ∈ C : |λ| <
| Arg(λ)| − απ/2
τ
α
, | Arg(λ)| >
απ
2
.
As illustrated by Figures 2.2-2.5, the sets Qτ
α(m) (m ∈ Z+
0 ) are disjoint and the
infinite union of their closures covers the whole complex plane. In the next two
section we detail the role of these sets in qualitative properties of (2.1).
Q (0)
Q (1)
Q (2)
Q (3)
ℑ( )
ℜ( )
Figure 2.2: Decomposition of eigenvalues’
complex plane for α = 0.4,
τ = 1.
ℜ
ℑ
α
τ
(1)α
τ
(2)α
τ
(3)α
τ (4)α
τ
Figure 2.3: Decomposition of eigenvalues’
complex plane for α = 1.1,
τ = 1
ℜ
ℑ
α
τ
(2)α
τ
(3)α
τ
(4)α
τ
Figure 2.4: Decomposition of eigenvalues’
complex plane for α = 2.1,
τ = 1
Figure 2.5: Decomposition of eigenvalues’
complex plane for α = 3.1,
τ = 1
Chapter 2. Analysis of one-term fractional delay differential systems 16
2.4 Asymptotically stable systems
The calculations behind the complex plane decomposition from the previous section
yield that the set Qτ
α(0) contains all the eigenvalues having solely characteristic roots
with negative real parts. Thus, it coincides with the stability region for (2.1) which
takes the form
Sτ
α = λ ∈ C : |λ| <
| Arg(λ)| − απ/2
τ
α
, | Arg(λ)| >
απ
2
. (2.7)
That enables us to write a fractional counterpart to Theorem 1.4 and simultaneously
a delay counterpart to Theorem 1.1 as follows
Theorem 2.9. Let A ∈ Rd×d
, τ ∈ R+
and α ∈ (0, 2). Then (2.1) is asymptotically
stable if and only if all eigenvalues λi (i = 1, . . . , d) of A are nonzero and satisfy
τ|λi|1/α
< | Arg(λi)| − απ/2 .
Moreover, if α /∈ Z+
, then the convergence to zero is of algebraic type; more precisely,
for any solution y of (2.1) there exists a suitable integer j ∈ {0, . . . , ⌈α⌉} such that
∥y(t)∥ ∼ tj−α−1
as t → ∞ (the symbol ∥ · ∥ means a norm in Rd
).
Proof. The proof is based on Theorems 2.2 and 2.6 combined with Lemma 2.7.
Its main challenge lies in asymptotic evaluation of the integral term
0
−τ
R(t − τ −
u)Aϕ(u)du. For details see [6,10,37].
Figures 2.6 and 2.7 illustrate evolution of the stability region for increasing α.
For all α ∈ (0, 1) ∪ (1, 2) the stability boundary has a cusp point at the origin from
which it continues symmetrically above and below real axis with tangents ±απ/2,
respectively. For α = 1, the cusp point smoothens as the tangents align with the
imaginary axis.
Figures 2.8 and 2.9 show the shape of the stability region for α close to integerorder
values one (compare to Figure 1.3) and two (the stability region vanishes).
Figure 2.10 outlines the effect of decreasing τ causing expansion of the stability
region up to the undelayed case for τ → 0+
(compare to Figures 1.1 and 1.2).
The most puzzling insight brought by changing α in (2.1) is depicted in Figure
2.11 where we see shape of stability region for the values α close to zero. Although for
all α > 0 the positive reals lie outside of stability region, we see that the limit shape
for α → 0+
tends to a circle. As shown in Figure 1.4, the circle is the known stability
region for difference equation (1.4) which, in a certain sense, can be seen as (2.1)
with α = 0. This remarkable connection suggests the potential of fractional-order
derivatives to provide transition not just between integer-order differential systems
but also between the differential and difference systems (for more comments see [6]).
The approach originating from Lemma 2.7 and Theorems 2.2 and 2.6 allows to
address also the boundary of stability region.
17 Chapter 2. Analysis of one-term fractional delay differential systems
 
ℜ
ℑ
− /2
/2 − /2
Figure 2.6: Stability region Sτ
α for
(2.1) depicted for α = 0.4, τ = 1.
ℜ
ℑ
− /2
2
2
Figure 2.7: Stability region Sτ
α for
(2.1) depicted for α = 1.1, τ = 1.
Theorem 2.10. Let A ∈ Rd×d
, τ ∈ R+
and α ∈ (0, 2). Then (2.1) is stable if and
only if all eigenvalues λ of A belong to Sτ
α or its boundary ∂Sτ
α, and all the ones
lying on the boundary have same algebraic and geometric multiplicities.
Proof. See [6,37].
Remark 2.11. (i) Comparing Theorems 2.9 and 2.10 we see that while presence of
an eigenvalue on the stability boundary removes asymptotic stability, it preserves
the stability provided it has the same same algebraic and geometric multiplicities.
Consequently, for scalar version of (2.1), the system is stable if and only if all
eigenvalues lie in the closure of Sτ
α (see also [39]).
(ii) Although this thesis deals with fractional derivatives of Caputo type, it is worth
noting that (2.1) with a Riemann-Liouville derivative has nearly the same stability
properties. The only difference occurs when the zero eigenvalue is present as proved
in [37]. Specifically, if α < 1 and the maximum size of any Jordan block associated
with the zero eigenvalue is less than 1/α, the asymptotic stability appears.
Oscillatory properties of asymptotically stable systems
Theorem 1.6 implies that in the case of the first-order delay system (1.3), the oscillations
occur for almost all λ (more precisely, some solutions of (1.3) do not oscillate,
if some eigenvalue lies in [−1/(τe), ∞), which is a set of zero measure). In the case
of α ̸= 1, the situation is very different. The following assertion shows that there
are no oscillatory solutions tending to zero.
Theorem 2.12. Let A ∈ Rd×d
, α ∈ R+
\ Z+
, τ ∈ R+
and let (2.1) be stable. If all
eigenvalues λ of A belong to Sτ
α ∪{0}, then all nonzero solutions are non-oscillatory.
Proof. The proof builds on Lemma 2.7 and the fact that in asymptotically stable case
the non-oscillating algebraic term Pλ,τ,m
α,β (t) dominates the oscillating exponential
Chapter 2. Analysis of one-term fractional delay differential systems 18
ℜ
ℑ
Figure 2.8: Stability region Sτ
α for
(2.1) depicted for α = 0.9, τ = 1 (the
corresponding limit case α → 1 is dot-
ted).
ℜ
ℑ
Figure 2.9: Stability region Sτ
α for
(2.1) depicted for α = 1.9, τ = 1 (the
corresponding limit case α → 2−
is an
empty set).
ℜ
ℑ
Figure 2.10: Stability region Sτ
α for
(2.1) depicted for α = 0.4, τ = 0.1
(the corresponding limit case τ → 0+
is dotted).
ℜ
ℑ
Figure 2.11: Stability region Sτ
α for
(2.1) depicted for α = 0.05, τ = 1
(the corresponding limit case α → 0+
is dotted).
functions. In case of the zero eigenvalue, the additional term is also non-oscillatory
no matter the multiplicity of the zero eigenvalue.
19 Chapter 2. Analysis of one-term fractional delay differential systems
2.5 Unstable systems
The techniques used in [6] to discuss properties of asymptotically stable systems turn
out to be effective also in the case of unstable system. In particular, they enable us
to describe the supremum asymptotics of the unbounded solutions as follows.
Theorem 2.13. Let A ∈ Rd×d
and α, τ ∈ R+
. Let λi be all distinct eigenvalues of
A (i = 1, . . . , n) and let (2.1) is not stable. Then solutions y(t) of (2.1) admit three
types of asymptotics:
(i) Let λ1 = 0 be the zero eigenvalue of A with algebraic multiplicity greater than
geometric one and let p1 be the maximal size of Jordan blocks corresponding to λ1.
Further, let λi ∈ Sτ
α for all i = 2, . . . , n. Then
∥y(t)∥ ∼ t(p1−1)α
as t → ∞ for any solution y(t) of (2.1).
(ii) Let λi (i = 1, . . . , ℓ ≤ n) be nonzero eigenvalues of A lying on ∂Sτ
α with algebraic
multiplicity greater than geometric one and let pi be the maximal size of Jordan
blocks corresponding to λi (i = 1, . . . , ℓ). Further, let λi ∈ Sτ
α for all i = ℓ + 1, . . . , n
provided ℓ < n and p = max(p1, . . . , pℓ). Then
∥y(t)∥ ∼sup tp−1
as t → ∞ for any solution y(t) of (2.1).
(iii) Let λi (i = 1, . . . , ℓ ≤ n) be eigenvalues of A located outside cl(Sτ
α) and let s1 be
the rightmost root of (2.5). Further, let λj, j ∈ L ⊂ {1, . . . , ℓ} be eigenvalues of A
such that (2.5) with λ = λj has the zero s1 and let p be the maximal size of Jordan
blocks corresponding to λj, j ∈ L. Then
∥y(t)∥ ∼sup tp−1
exp{ℜ(s1)t} as t → ∞ for any solution y(t) of (2.1).
Proof. The details see in [10].
To obtain an actually effective (and non-improvable) asymptotic result for the
solutions of (2.1), we have to look at the problem inversely. More precisely, for
a given complex λ /∈ Sτ
α, we need to find (nonnegative) real values u, v such that
the rightmost root s1 of (2.5) satisfies ℜ(s1) = u, |ℑ(s1)| = v.
This nontrivial question can be addressed using properties and methods of Lambert
function, i.e. the function introduced as a solution of W(z) exp(W(z)) = z,
z ∈ C (see, e.g. [28]). In [15] we developed a framework allowing to evaluate the
precise asymptotic envelop of the unbounded solutions and also the corresponding
asymptotic frequency of oscillations. For the sake of simplicity, only scalar version
of (2.1) with complex coefficient λ was considered. The findings can be summarized
in the following
Theorem 2.14. Let α ∈ (1, ∞), τ ∈ R+
and λ ∈ C. If λ /∈ Sτ
α, then, for any
solution y(t) of Dα
0 y(t) = λy(t − τ) it holds
y(t) = exp(ut)(c exp(i vt) + o(1)) as t → ∞
Chapter 2. Analysis of one-term fractional delay differential systems 20
where c is a complex constant, u ≥ 0 is the unique solution of
α arccos
u exp(τu/α)
|λ|1/α
+
τ |λ|2/α − u2 exp(2τu/α)
exp(τu/α)
= |Arg(λ)| ,
v > 0 is the unique solution of
vα
sinα
(|Arg(λ)| − τv)/α
exp τv cot (|Arg(λ)| − τv)/α = |λ|, if |Arg(λ)| > 0 ,
and v = 0 if Arg(λ) = 0.
Proof. The first and simple part is to express the characteristic roots sk (k ∈ Z) of
(2.5) in terms of Lambert function, i.e.
sk =
α
τ
Wk
τ
α
λ1/α
, k ∈ Z ,
where Wk is the kth branch of the Lambert function. The key part is to prove the
existence of ordering put on Lambert functions branches, namely that ℑ(Wk(z)) ≤
ℑ(Wk+1(z)) for all k ∈ Z and z ̸= 0 and ℜ(W0(z)) ≥ ℜ(Wk(z)) for all k ∈ Z.
The proof is then concluded by technically challenging calculations leading to the
equations for u, v depending on α, τ, λ. For details we refer to [15].
Remark 2.15. (i) Note that Theorem 2.14 is not formulated for α < 1. That is
a consequence of 1/α occurring in the argument of the Lambert function which for
α < 1 might introduce some additional roots which do not actually solve (2.5). This
problem does not seem to be solvable in the framework of Lambert function method
and, to the author’s knowledge, remains open.
(ii) Figure 2.12 depicts a practical "map" allowing us to quickly estimate asymptotic
modulus and oscillation frequency for the solution of Dα
0 y(t) = λy(t − τ) based on
location of λ in the complex plane.
(iii) Theorem 2.14 considers only scalar version of (2.1) with complex coefficient. If
we deal with the vector version of (2.1) and the system matrix A has eigenvalues
with the same algebraic and geometric multiplicities, we just have to apply Theorem
2.14 for every eigenvalue and combine the results (Figure 2.12 also applies). In
case of different algebraic and geometric multiplicities, the estimates for asymptotic
frequencies are still valid and the estimates for the modulus have to adjusted by
polynomial multiplication.
Oscillatory properties of unstable systems
As in the stable case, (2.1) (for α ̸= 1) does not have any combination of entry
parameters guaranteeing oscillations of all solutions. On the other hand, there are
combinations that ensure no oscillatory solutions.
Theorem 2.16. Let A ∈ Rd×d
, α ∈ R+
\ Z+
, τ ∈ R+
and let (2.1) be unstable. If
all eigenvalues λ of A belong to Sτ
α ∪{0}∪(Q1(α, τ)∩R), then all nonzero solutions
are non-oscillatory.
21 Chapter 2. Analysis of one-term fractional delay differential systems
1.25<| ( )|<1.501
( )=0.251
0.50
0.50
0.25
0.75
0.75
1.00
1.00
1.25
1.50
1.75
2.00
( )=1.251
( )<1.0010.75<
-0.25<| ( )|<0.251
( )<0.7510.50<
Figure 2.12: The blue curves represent the set of all λ ∈ C such that the
rightmost characteristic root s1 of (2.5) satisfies ℑ(s1) = v, and the particular
orange curves represent the set of all λ ∈ C such that the rightmost
characteristic root s1 of (2.5) satisfies |ℜ(s1)| = v (the scenario corresponds
to α = 1.2 and τ = 1). As an example, there are highlighted curvilinear
rectangles representing sets of all λ ∈ C yielding ℜ(s1) and ℑ(s1) from a
certain range.
Proof. The outline of the prove is following, for details we refer to [10].
It can be seen from Lemma 2.7 that oscillatory solutions can occur only if there
is a positive real characteristic root. Further, it is possible to prove that (2.5) has
a positive real root if and only if λ is a positive real and this root is simple, unique
and it is the rightmost root of (2.5).
Then we employ properties of Qτ
α(m) introduced in Section 2.3. In particular,
that there exist just m (m = 0, 1, . . . ) characteristic roots of (2.5) with a positive
real part (while remaining roots have negative real parts) if and only if λ ∈ Qτ
α(m).
Moreover, (2.5) has a root with the zero real part if λ ∈ ∂[Qτ
α(m)] for some m =
0, 1, . . . .
Remark 2.17. Similarly to Theorems 2.12 and 2.16, oscillatory solutions can occur
in some cases only for a particular choice of initial conditions (see [10]).
Chapter 2. Analysis of one-term fractional delay differential systems 22
Chapter 3
Analysis of two-term fractional delay
differential equations
This chapter summarizes the key findings related to two-term FDDE from author’s
papers [11–13]. Building on our analysis of one-term FDDS, it would seem natural
to turn to
Dα
0 y(t) = Ay(t) + By(t − τ) ,
where A, B are real d × d matrices and τ is a positive real time delay. Moreover,
such a mathematical model would provide a large application potential (see, e.g.
[40,43]), especially in control theory regarding stabilization of equilibria of fractional
dynamical systems via delayed feedback controls. Although addition of Ay(t) on the
right-hand side looks quite straightforward, it highly increases the difficulty of the
studied problem. Even the classical case α = 1 is still generally unsolved (see,
e.g. [4,31,46,55]).
That is why we focus on the proper development of stability theory for the scalar
case, namely two-term fractional delay differential equation (FDDE)
Dα
0 y(t) = ay(t) + by(t − τ) , (3.1)
where a, b are real coefficients, τ > 0 is a real lag and α ∈ (0, 2). Similarly as for
(2.1), the associated initial conditions are considered as
y(t) = ϕ(t) , t ∈ [−τ, 0) , (3.2)
lim
t→0+
y(j)
(t) = ϕj , j = 0, 1 (3.3)
where ϕ is absolutely Riemann integrable on [−τ, 0) and ϕj are reals.
The topic of stability and asymptotic analysis of FDDEs attracts the attention
of many authors. Before 2016, significant majority of corresponding stability results
was derived as parametric equations or implicit relations for the stability boundary
or usually as an outcome of the D-decomposition method and an appropriate root
locus. For such or similar stability results on (3.1) (with α ∈ (0, 1)) we refer to [1,30],
but the trend is evident from the literature even for simpler cases (see, e.g. [16,40,
– 23 –
Chapter 3. Analysis of two-term fractional delay differential equations 24
41,54]). Hence, we decided to focus on the formulation of explicit stability criteria
for (3.1), as they provide much more accessible and practical tool in comparison to
the usual parametric or implicit ones. In [13] we succeeded for the derivative order
less than one and managed to find the explicit description of stability region in the
(a, b)-plane and the formula for the change from stability to instability with respect
to increasing τ which is present also in the first-order case.
It is well-known that the integer-order linear delay dynamical systems may
change their stability into instability with growing time delay not just once, but
repeatedly back and forth. This interesting phenomenon, often referred to as stability
switching, is still a current subject of research as exemplified, e.g. by [19,42,47,51]
where values of stability switches are described via parameters of the corresponding
integer-order system. Thus, the occurrence of stability switching for FDDEs is a
natural topic discussed in the second decade of 21st century, e.g. in [56, 57]. Our
papers [11,12] are mostly devoted to this area, the former one considering (3.1) with
imaginary coefficient a and α less than one, and the latter one with α between one
and two. In both the cases we managed to derive explicit values of stability switches
as well as conditions for so-called delay-independent stability. Moreover, [12] clarifies
the continuous transition between qualitatively very different stability regions
for (3.1) with α = 1 and α = 2.
The following sections are built on the main results of [11–13]. Section 3.1 elaborates
on structure and asymptotics of solutions to (3.1). Then, unlike the Chapter 2,
we focus less on precision of asymptotics and more on shape of stability regions and
their dependence on system parameters. Section 3.2 deals with the case of derivative
order less than one, Section 3.3 changes one of the system parameters from real to
imaginary and Section 3.4 comes back to real (3.1) with order between one and two.
Throughout the chapter, we put stress on the explicit stability conditions which are
quite challenging for (3.1), among other things, because of the presence of stability
switches (see Sections 3.3 and 3.4).
3.1 Structure and asymptotics of solutions
Because (3.1) shares with (2.1) its linearity, fractional derivative order as well as
time delay, the structure of the solution is expected to be similar. The Laplace
transform of (3.1)-(3.3) shows that the fundamental solution belongs to the family
of functions
Ra,b,τ
α,β (t) = L−1 sα−β
sα − a − b exp[−sτ]
(t) (3.4)
where α, β, τ > 0 and a, b ∈ R. We can see that the generalized delay exponential
function (with parameter m = 0) introduced by Definition 2.3 in the previous
chapter is the special case of (3.4). Indeed, R0,b,τ
α,β (t) = Gb,τ,0
α,β (t) (see (2.6)).
Using the inverse Laplace transform we arrive at the representation of the solu-
25 Chapter 3. Analysis of two-term fractional delay differential equations
tion to (3.1)-(3.3) (compare to Theorem 2.2)
y(t) =
⌈α⌉
j=0
ϕjRa,b,τ
α,j+1(t) + b
0
−τ
Ra,b,τ
α,α (t − τ − u)ϕ(u)du . (3.5)
The characteristic equation associated with (3.1) is implied by (3.4) and (3.5) in
the expected form
sα
− a − b exp(−sτ) = 0 (3.6)
which has infinitely many complex roots (compare to (2.5) and to characteristic
equations for the classical integer-order cases in Section 1.3).
The key auxiliary assertion, the counterpart to Lemma 2.7, deals with asymptotic
properties of Ra,b,τ
α,β functions.
Lemma 3.1. Let α ∈ (0, 1), β ∈ (0, 1], a, b ∈ R and τ ∈ R+
and let si be roots of
(3.6).
(i) If ℜ(si) < 0 for all si then
Ra,b,τ
α,β (t) ∼ tβ−α−1
for α ̸= β and Ra,b,τ
α,α (t) = O(t−α−1
) as t → ∞ .
(ii) If there exists the zero root of (3.6) and ℜ(si) < 0 otherwise, then
Ra,b,τ
α,1 (t) ∼ 1 and Ra,b,τ
α,β (t) = O(tβ−1
) for β < 1 as t → ∞ .
(iii) If ℜ(si) ≤ 0 for all si and some of the roots are purely imaginary, then
Ra,b,τ
α,β (t) ∼sup 1 as t → ∞ .
(iv) If ℜ(si) > 0 for some si then
Ra,b,τ
α,β (t) ∼sup (Bt + C) exp[Mt] as t → ∞
where M = maxsi
(ℜ(si)) and reals B, C ≥ 0 are such that B + C > 0.
Proof. The proof utilizes the technique already outlined in the proof of Lemma 2.7,
with much higher technical difficulty, more branching to be considered with respect
to the parameter values and with several adjustments (see [13, pages 344–349]).
The assumptions for the use of the technique needs the root analyses of (3.6),
most importantly showing that for an arbitrary 0 < ω < π, the characteristic
equation has no more than a finite number of roots s such that | Arg(s)| ≤ ω.
Although Lemma 3.1 is formulated for real values of a, b and α less than one, it
is only a technical matter to generalize it. The quality of the asymptotic estimates
may be affected but the stability implications remains the same (as pointed out
in [11,12]).
Theorem 3.2. Let α > 0, τ > 0 and a, b be complex numbers.
(i) If all the roots of (3.6) have negative real parts, then (3.1) is asymptotically
stable.
(ii) If there exists a root of (3.6) with a positive real part, then (3.1) is not stable.
Remark 3.3. Theorem 3.2 does not address the stability boundary. As we will
discuss later, the stability boundary for (3.1) contains points of asymptotic stability,
stability and also instability for various values of system parameters.
Chapter 3. Analysis of two-term fractional delay differential equations 26
3.2 Stability regions for orders less than one
Let us consider (3.1) with α < 1 and investigate the boundary locus for the corresponding
(3.6), i.e. the set of all real couples (a, b) such that the characteristic
equation admits a root with zero real part. Substituting s = ± i φ into (3.6) and considering
real and imaginary parts separately yields two qualitatively distinct parts of
boundary locus: the line a + b = 0 (corresponding to the zero root) and the system
of curves (corresponding to purely imaginary roots)
am(ρ) =
ρα
sin(ρ + απ/2)
τα sin(ρ)
, bm(ρ) = −
ρα
sin(απ/2)
τα sin(ρ)
, (3.7)
mπ < ρ < (m + 1)π, m = 0, 1, . . . . The curves forming the boundary locus are
depicted in the (a, b)-plane on Figure 3.1 (see also [30] where the authors redundantly
considered multi-valued function sα
instead of the single-valued one).
a + b = 0
m = 1
m = 0
m = 2
m = 4
a
b
m = 3
m = 5
Figure 3.1: Boundary locus of (3.1) for α = 0.4, τ = 1.
The necessary link between the boundary locus curves and stability properties
of (3.1) is provided by the following
Theorem 3.4. Let 0 < α < 1, a, b and τ > 0 be real numbers. Then all roots of
(3.6) have negative real parts if and only if the couple (a, b) is an interior point of
the area bounded by the line a + b = 0 from above and by the parametric curve
a =
ρα
sin(ρ + απ/2)
τα sin(ρ)
, b = −
ρα
sin(απ/2)
τα sin(ρ)
, ρ ∈ ((1 − α)π, π) (3.8)
from below.
27 Chapter 3. Analysis of two-term fractional delay differential equations
Proof. It follows from continuous dependence of roots of (3.6) on the coefficients a,
b. This property particularly implies that the number of characteristic roots with a
positive real part remains unchanged in all open sets whose boundaries are formed by
the line a+b = 0 or by some curves (3.7). Then it is enough to choose representatives
of these open sets to specify the number of roots of (3.6) with positive real parts
within these sets. For details see [13].
Remark 3.5. Theorem 3.4 implies that of all curves in the system (3.7) only a part
of the curve characterized by m = 0 affects the stability boundary (see Figure 3.2).
Others lie in the region where (3.1) is not stable. Also, substituting (1 − α)π into
(3.8) enables us to calculate the coordinates of the cusp point (see also Figure 3.2).
[ (1 - )]
2 cos( /2)
[ (1 - )]
2 cos( /2)
 
;
b = a - ( / ) cos( /2)
a
b
Figure 3.2: The stability region Sτ
α for (3.1) depicted for α = 0.4 and τ = 1.
The main objective of our effort is to derive explicit conditions determining the
stability region. A related problem has been discussed in [1] where (3.6) is analysed
for a < 0. Here we present the assertion removing the restriction on a and yielding
results in a simpler form due to the use of a different computational technique.
Theorem 3.6. Let 0 < α < 1, a, b and τ > 0 be real numbers. Then (3.1) is
asymptotically stable if and only if it holds either
a ≤ b < −a and τ is arbitrary , (3.9)
or
|a| + b < 0 and τ < τ∗
=
(1 − α)π/2 + arccos[(−a/b) sin(απ/2)]
[a cos(απ/2) + (b2 − a2 sin2
(απ/2))1/2]1/α
. (3.10)
Chapter 3. Analysis of two-term fractional delay differential equations 28
Proof. The proof is based on rewriting the parametric equations (3.8) into the explicit
ones via intersection analysis of the boundary locus curves, elimination of the
parameter φ and careful operations with inverse trigonometric functions. For details
see [13].
Remark 3.7. (i) Clearly, the stability region Sτ
α consists of pairs (a, b) such that either
(3.9) or (3.10) holds. The condition (3.9) defines the region of delay-independent
stability. The second condition, (3.10), shows dependence on the time delay, namely
it indicates the one-time loss of stability with increasing τ reaching the value τ∗
.
(ii) The delay-dependent part of stability region expands with decreasing time delay.
If we consider the limit τ → 0+
, the stability region simplifies into half-plane
a + b < 0. That agrees with the stability region for the scalar version of (1.2) with
the coefficient a + b which is the limit of (3.1) for τ → 0+
.
(iii) Considering (3.1) with a = 0, we obtain the scalar version of (2.1) with coefficient
b which is asymptotically stable for −(π/τ −απ/(2τ))α
< b < 0 (see Figures 2.6
and 2.7). That corresponds to Theorem 3.6 as −(π/τ −απ/(2τ))α
is the intersection
between b-axis and the lower branch of the stability boundary.
The following theorem shows that for α less than one, the stability boundary
fully corresponds to the situation when (3.1) is stable but not asymptotically stable.
Theorem 3.8. Let 0 < α < 1, a, b and τ > 0 be real numbers. Then (3.1) is stable,
but not asymptotically stable, if and only if either
a + b = 0, a ≤
[π(1 − α)]α
2τα cos(απ/2)
, (3.11)
or
|a| + b < 0, τ = τ∗
, τ∗
being the same expression as in (3.10). (3.12)
Proof. The assertion follows from application of Lemma 3.1, see [13].
Remark 3.9. The stability of (3.1) in the cusp point (the intersection of the line
a + b = 0 and (3.8) is not analogue to the situation known from (1.5). It can be
proved by a direct calculation that (1.5) at the cusp point (i.e. a = −b = 1/τ) is
not stable.
Due to the scalar nature of (3.1) and quite simple form of the stability boundary
for of α less than one, we have quite comprehensive asymptotic description of
solutions.
Lemma 3.10. Let 0 < α < 1, a, b and τ > 0 be real numbers and let y be a solution
of (3.1).
(i) Let (3.1) be asymptotically stable. Then
y(t) ∼ t−α
or y(t) = O(t−α−1
) as t → ∞ .
29 Chapter 3. Analysis of two-term fractional delay differential equations
(ii) Let (3.1) be stable but not asymptotically stable. If (3.11) is satisfied, then
y(t) ∼ 1 or y(t) = O(tα−1
) as t → ∞ .
If (3.12) holds, then
y(t) ∼sup 1 or y(t) = O(1) as t → ∞ .
(iii) Let (3.1) be unstable. Then y(t) = O(t exp[Mt]) as t → ∞, where M =
maxsi
(R(si)), si being roots of (3.6). Moreover, there exists a solution y of (3.1)
such that
y(t) ∼sup t exp[Mt] or y(t) ∼sup exp[Mt] as t → ∞ .
Proof. The proof is a consequence of Lemma 3.1 and (3.5). See [13] for details.
Remark 3.11. As usual for asymptotically stable fractional dynamic systems, the
decay rate of the solutions is algebraic, while in the classical case (1.5) it is known
to be exponential.
3.3 Stability regions for orders less than one and an
imaginary coefficient
Let us change the first coefficient of (3.1) into an imaginary one and study the
problem
Dα
0 y(t) = i ay(t) + by(t − τ) . (3.13)
Although (3.13) might look artificially constructed, it actually plays a key role in
the stability investigation of a planar fractional delay system
Dα
0 x1(t) = ux1(t − τ) + vx2(t)
Dα
0 x2(t) = wx1(t) + ux2(t − τ)
with real entries u, v, w (v, w ̸= 0) which was the focus of [11]. We note that the
corresponding classical case (α = 1) was studied by [46] due to its stability switching
nature.
Finding the boundary locus for the characteristic equation associated with (3.13),
sα
− i a − b exp(sτ) = 0 , (3.14)
starts in the same manner as in the previous section but soon key differences appear.
First, (3.14) does not admit zero root. Second, purely imaginary roots induce the
system of curves in a form
am(ρ) = ±
ρα
sin(ρ + απ/2)
τα cos(ρ)
, bm(ρ) =
ρα
cos(απ/2)
τα cos(ρ)
, (3.15)
Chapter 3. Analysis of two-term fractional delay differential equations 30
0 < ρ < π/2 for m = 0 and mπ − π/2 < ρ < mπ + π/2 for m ∈ Z+
while
ρ + απ/2 ̸= mπ for m ∈ Z+
0 . Even though (3.15) looks formally similar to (3.7), the
description of the stability boundary is now much more complicated as it is formed
by parts of all curves (3.15) requiring calculations of infinitely many intersections.
For the precise procedure we refer to [11] and state the end result:
Let us define two curves
Γ+
=
∞
m=0
Γ2m and Γ−
=
∞
m=0
Γ2m+1
composed of the system of curves Γm in the (a, b)-plane given by (3.15) with the
parameter restriction
(2m + 1 − α)π
2
− θ∗
m−2 < ρ <
(2m + 1 − α)π
2
+ θ∗
m, m ∈ Z+
0
where θ∗
m ∈ (0, απ/2) is the unique root of
−
sin(θ + απ/2)
sin(θ − απ/2)
=
(2m + 3 − α)π
θ + (m + 1/2 − α/2)π
− 1
α
, m ∈ Z+
0 (3.16)
and θ∗
−2 = (α − 1)π/2, θ∗
−1 = π/2. Note that in the first relation of (3.15), both the
sign cases have to be considered, and thus any curve Γm has two branches symmetric
with respect to b-axis.
Using this notation we can write
Lemma 3.12. Let 0 < α < 1, τ > 0, a ̸= 0 and b be real numbers. Then (3.13) is
asymptotically stable if and only if the couple (a, b) is located inside the area bounded
by Γ+
from above and by Γ−
from below.
Proof. The proof is using the connection between the growth of |b| and number of
characteristic roots with a positive real part, and analysis of intersections among Γm.
Moreover, it employs appropriate asymptotic properties of the functions R± i a,b,τ
α,α ,
R± i a,b,τ
α,1 which, as a by-product, also implies the algebraic decay rate of solutions.
For details see [11].
Remark 3.13. (i) Lemma 3.12 describes parametrically the stability region Sτ
α of
(3.13). Figure 3.3 depicts the known result for α = 1 (see [47]) and Figures 3.4, 3.5
and 3.6 show the evolution of the stability region for decreasing derivative order.
They illustrate that region of delay-independent stability occurs for any α < 1.
(ii) If we consider lines connecting the origin and cusp points of Γ+
, we can prove
that they have decreasing tangents with respect to increasing index of a given cusp
point starting closest to b-axis (a similar comment is true also for Γ−
). This fact
plays a central role in the context of stability switching.
(iii) Γ+
has the tangent ± cot(απ/2) at the origin. Let us consider a line connecting
31 Chapter 3. Analysis of two-term fractional delay differential equations
a
b
Figure 3.3: A classical result for the
stability region Sτ
1 with τ = 1.
Γ+
Γ-
b
a
Figure 3.4: The stability region Sτ
α for
α = 0.95, τ = 1.
Γ+
Γ-
a
b
Figure 3.5: The stability region Sτ
α for
α = 0.6, τ = 1.
Γ+
Γ-
a
b
Figure 3.6: The stability region Sτ
α for
α = 0.2, τ = 1.
origin with the first cusp point of Γ+
(i.e. the endpoint of Γ0). It can be calculated
that the tangent of this line is equal to ± cot(απ/2) if and only if α = α∗
where
α∗
≈ 0.6150768144 . (3.17)
If α > α∗
, then for some values of b/a the first stability switch changes instability into
stability. If α < α∗
, the first stability switch is always from stability to instability.
For the full calculation we refer to [11].
The shape of the stability boundary for (3.13) is quite complex and its proper
reformulation into the explicit form of stability conditions brings many challenges.
Chapter 3. Analysis of two-term fractional delay differential equations 32
It seems that the most useful perspective is provided by considering the ratio b/a
as presented in the main result of this section:
Theorem 3.14. Let 0 < α < 1, τ > 0 and a, b be real numbers, let α∗
be given by
(3.17) and let θ∗
m ∈ (0, απ/2) be the unique root of (3.16) for m ∈ Z+
0 . Further,
assuming |b|/|a| ≥ cos(απ/2), let n ≥ 0 be an even integer (if b ≥ 0), or an odd
integer (if b < 0), uniquely determined by
cos(απ/2)
cos(θ∗
n)
≤
|b|
|a|
<
cos(απ/2)
cos(θ∗
n−2)
, n ≥ 2 , or
cos(απ/2)
cos(θ∗
n)
≤
|b|
|a|
, n ∈ {0, 1} .
(3.18)
The zero solution to (3.13) is asymptotically stable if and only if any of the following
conditions holds:
|b|
|a|
< cos(απ/2) ; (3.19)
cos(απ/2) ≤
b
|a|
< cot(απ/2) and τ ∈
n/2−1
j=−1
(τ∗
2j,−1, τ∗
2j+2,1) ; (3.20)
α > α∗
, cot(απ/2) ≤
b
|a|
<
cos(απ/2)
cos(θ∗
0)
and τ ∈
n/2−1
j=0
(τ∗
2j,−1, τ∗
2j+2,1) ; (3.21)
b
|a|
≤ − cos(απ/2) and τ ∈
(n−1)/2−1
j=−1
(τ∗
2j+1,−1, τ∗
2j+3,1) (3.22)
where τ∗
i,κ = 0 for negative integers i and τ∗
i,κ = τ∗
i,κ(a, b) where i ∈ Z+
0 , κ = ±1 and
τ∗
i,κ(a, b) =
(i + (1 − α)/2) π − κ arccos (|a/b| cos(απ/2))
κ b2 − a2 cos2(απ/2) + |a| sin(απ/2)
1/α
.
Proof. The proof of this theorem requires many preliminary assertions such as guaranteeing
existence, uniqueness and ordering of θ∗
m (m ∈ Z+
0 ), derivation of α∗
and
analysis of the ratio b/a for points belonging to Γ+
and Γ−
. In particular, for a
given ratio |b/a|, an analytical description of delays τ such that (a, b) ∈ Γ+
∪Γ−
has
to be given. For details see [11, pages 7-13].
Remark 3.15. (i) We note that the nonlinear inequalities (3.18) enable us to determine
the exact number of stability switches. For (3.20) then there are n+1 switches
(odd number), for (3.21) there are n switches (even number) and for (3.22) there
are n switches (odd number).
(ii) The condition (3.19) describes the region of delay-independent stability symmetric
with respect to both axes. Decreasing α expands this region towards the cone
b = |a| which is the limit case for α → 0+
.
(iii) For b/|a| > 0 we can see in Figures 3.8, 3.9 and 3.10 the role of the value α = α∗
.
It separates two qualitatively different patterns of stability switching, namely for
α ≤ α∗
it always starts with the first switch changing stability into instability.
33 Chapter 3. Analysis of two-term fractional delay differential equations
α = 1
τa
b/|a|
Figure 3.7: A classical result for the
stability region Sτ
1 (α = 1) in the
(τa, b/|a|)-plane, see [46].
τα
a
b/|a| α > α*
Figure 3.8: The stability region Sτ
α in
the (τα
a, b/|a|)-plane for α = 0.8 (i.e.
α > α∗
).
τα
a
b/|a| α = α*
Figure 3.9: The stability region Sτ
α in
the (τα
a, b/|a|)-plane for α = α∗
.
τα
a
b/|a| α < α*
Figure 3.10: The stability region Sτ
α
in (τα
a, b/|a|)-plane for α = 0.4 < α∗
.
3.4 Stability regions for orders from one to two
Let us consider (3.1) with α ∈ (1, 2). The boundary locus formulas have the same
form as for the case α ∈ (0, 1), i.e. it is formed by the line a + b = 0 and the system
of curves Γm in the (a, b)-plane given by (3.7). Properties of these curves for higher
α significantly differ from the case α ∈ (0, 1) and are actually qualitatively more
similar to (3.15) discussed in the previous section.
In [12] it proved to be useful for lucidity, to think of this system of curves from
Chapter 3. Analysis of two-term fractional delay differential equations 34
two perspectives: their asymptotes and intersections.
Lemma 3.16. Let α ∈ (1, 2), τ ∈ R+
and let Γm = {(a, b) ∈ R2
: a = am(ρ), b =
bm(ρ), ρ ∈ (mπ, mπ + π)} (m = 0, 1 . . . ) be the curves defined by (3.7). Then it
holds:
(i) The line a + b = 0 is tangent to the curve Γ0 at the origin, and the line
p−
0 : b = a −
π
τ
α
cos
απ
2
is the asymptote to Γ0 as ρ → π−
. Moreover, b0(ρ) < a0(ρ) − (π/τ)α
cos(απ/2),
b0(ρ) < 0 and b0(ρ) < −a0(ρ) for all ρ ∈ (0, π).
(ii) If m is a positive odd integer, then Γm has asymptotes p+
m (as ρ → mπ+
) and
p−
m (as ρ → (m + 1)π−
) given by
p+
m : b = a −
mπ
τ
α
cos
απ
2
and p−
m : b = −a +
mπ + π
τ
α
cos
απ
2
.
Moreover, bm(ρ) > 0, bm(ρ) > am(ρ) − (mπ/τ)α
cos(απ/2) and bm(ρ) > −am(ρ) +
((mπ + π)/τ)α
cos(απ/2) for all ρ ∈ (mπ, (m + 1)π).
(iii) If m is a positive even integer, then Γm has asymptotes p+
m (as ρ → mπ+
) and
p−
m (as ρ → (m + 1)π−
) given by
p+
m : b = −a +
mπ
τ
α
cos
απ
2
and p−
m : b = a −
mπ + π
τ
α
cos
απ
2
.
Moreover, bm(ρ) < 0, bm(ρ) < −am(ρ) + (mπ/τ)α
cos(απ/2) and bm(ρ) < am(ρ) −
((mπ + π)/τ)α
cos(απ/2) for all ρ ∈ (mπ, (m + 1)π).
Proof. The proof is of a technical nature utilizing limits calculations for (3.7). For
details see [12].
Lemma 3.17. Let α ∈ (1, 2), τ ∈ R+
and let et Γm = {(a, b) ∈ R2
: a = am(ρ), b =
bm(ρ), ρ ∈ (mπ, mπ + π)} (m = 0, 1 . . . ) be the curves defined by (3.7). Further, let
Xm,n = (am,n, bm,n) be intersections of Γm and Γn (if they exist). Then it holds:
(i) The intersection (am,n, bm,n) exists (and it is unique) if and only if m, n have the
same parity.
(ii) am,m+2k < 0 for all k ∈ Z such that k > −m/2.
(iii) am,m+2k > am,m+2(k+1) for all k ∈ Z such that k > −m/2.
(iv) am,m+2k > am+2ℓ,m+2k+2ℓ for all k ∈ Z such that k > −m/2 and ℓ = 1, 2 . . . .
Proof. The question of analysing intersections of Γm, Γn is transformed into root
study of an equation involving transcendental expression similar to (3.16). For detail
see [12].
Remark 3.18. (i) Lemma 3.16 says that each curve Γm (m = 0, 1 . . . ) is contained
in an infinite trapezoid consisting of its asymptotes and the a-axis. Each pair Γm,
Γm+1 shares a common asymptote as depicted in Figure 3.11.
35 Chapter 3. Analysis of two-term fractional delay differential equations
2
a
1
b
p1
+
≡ p2
+
p1
_
p2
_
Figure 3.11: The common asymptote
to Γ1 and Γ2 and the corresponding
trapezoids for α = 1.8 and τ = 1.
3
5
7
1
2
4
6
0
b
a
X0,2X2,4
X4,6
X2,6
X0,6
X0,4
X1,3
X3,5
X1,5
X3,7
Figure 3.12: Some intersections Xm,n
for α = 1.8, τ = 1 and m, n ∈
{0, 1, 2, 3, 4, 5, 6, 7}.
(ii) Figure 3.12 demonstrates the locations and ordering of intersections Xm,n described
in Lemma 3.17.
(iii) A similar asymptotes and intersections analyses might be useful also in the case
of (3.15), however it was not the point of study in [11].
In order to describe the stability region, we introduce the following notation. Let
P be the line segment
a = −ρ, b = ρ, ρ ∈ 0,
(3π − απ)α
2τα| cos(απ/2)|
and let ˜Γm (m = 0, 1, . . . ) be the parts of Γm with the endpoints Xm,m−2 and Xm,m+2
given by its intersections with the neighbouring curves Γm−2, Γm+2 (see Figure 3.12),
by origin for m = 0 and by the second endpoint of P for m = 1. Further, we put
ΓAS
=
∞
m=0
˜Γm ∪ P .
Using this notation, the geometric description of the stability region is provided by
the following assertion (compare to Theorem 3.4 and Lemma 3.12).
Theorem 3.19. Let α ∈ (1, 2), τ ∈ R+
and a, b ∈ R. Then (3.1) is asymptotically
stable, if the couple (a, b) is located inside the area containing the negative part of
a-axis and bounded by ΓAS
.
Moreover, (3.1) is not stable, if (a, b) lies inside the area containing the positive part
of a-axis and bounded by ΓAS
.
Chapter 3. Analysis of two-term fractional delay differential equations 36
a
a
b
b
AS
Γ
b=-a
Sα
απ
b=-sin( )a
απ
2b=sin( )a
τ
Figure 3.13: Stability boundary ΓAS
and stability region Sτ
α of (3.1) for α =
1.8 and τ = 1.
a
b
b=-a
AS
Γ
απ
b=sin( )a
Sα
a
b
1
1
απ
b=-sin( )a
τ
Figure 3.14: Stability boundary ΓAS
and stability region Sτ
α of (3.1) for α =
1.4 and τ = 1.
Proof. In order to complete this proof, the preconditions similar to the case of
α ∈ (0, 1) have to validated and recalculated (see proofs of Theorem 3.4 and Lemma
2.7). For details we refer to [12] and highlight here only the most interesting partial
results such as:
If (a, b) ∈ Γm (for unique m) and |b| increases, then a new characteristic root
with a positive real part appears.
If a < 0 and b ∈ R, then there exists δ = δ(α, a) > 0 such that all characteristic
roots have negative real parts whenever |b| < δ.
The nonzero characteristic roots depend on a, b continuously.
The case α ∈ (1, 2) stands out mainly due to the need to consider the occurrence
of multiple roots. We proved that a characteristic root has multiplicity greater than
one if and only if either it is zero or there exists an integer k such that αρ − ρ +
τr sin(ρ) = kπ and τr sin(αρ)+α sin(αρ−ρ) = 0. Moreover, any characteristic root
has multiplicity at most three.
Remark 3.20. (i) Although Theorem 3.19 gives only sufficient conditions for asymptotic
stability, its second part makes them near-optimal. The only additional points
where the stability might occur, lie on the stability boundary given by ΓAS
.
(ii) The stability region Sτ
α described in Theorem 3.19 is depicted in Figures 3.13
and 3.14 including a detail of the situation near the origin. Comparing these details
to Figure 1.5 suggests the limit transition for α → 1+
, as the rightmost point of Γ1
changes into the cusp point appearing for α ≤ 1.
Our main goal is to obtain the explicit stability conditions, not just a geometric
description of the stability boundary as in Theorem 3.19. In the sequel, we provide
an alternative stability criterion for the case a < 0 that better agrees with the
37 Chapter 3. Analysis of two-term fractional delay differential equations
form of the conditions of Theorems 1.7 and 1.9. We do not consider the case a > 0
because the corresponding stability conditions are quite straightforward (see Figures
3.13 and 3.14).
Theorem 3.21. Let α ∈ (1, 2), τ > 0, a < 0 and b be real numbers and τ+
ℓ , τ−
ℓ be
defined as
τ±
ℓ =
(ℓ + 1∓1
2
)π + (2−α)π
2
± arcsin a
b
sin(απ
2
)
a cos(απ
2
) ± b2 − a2 sin2
(απ
2
)
1/α
, ℓ ∈ Z+
0 .
(i) If − sin(απ/2) < b/a < sin(απ/2), then (3.1) is asymptotically stable.
(ii) If b/a > sin(απ/2), then there exists an integer N1 ≥ 0 such that (3.1) is asymptotically
stable for any τ ∈ (τ−
2k−2, τ+
2k), and it is not stable for any τ ∈ (τ+
2k, τ−
2k+2)
where k = 0, . . . , N1 (here we set τ−
−2 = 0, τ−
2N1+2 = ∞).
(iii) If −1 < b/a < − sin(απ/2), then there exists an integer N2 ≥ 0 such that
(3.1) is asymptotically stable for any τ ∈ (τ−
2k−1, τ+
2k+1), and it is not stable for any
τ ∈ (τ+
2k+1, τ−
2k+3) where k = 0, . . . , N2 (here we set τ−
−1 = 0, τ−
2N2+3 = ∞).
(iv) If b/a < −1, then (3.1) is not stable.
Proof. See [12].
Remark 3.22. (i) Comparing to Theorems 3.21 and 3.14 we notice two obvious
differences: In the real case (Theorem 3.21), there is no special value α∗
changing
the switching pattern, and the real case is not symmetric with respect to b-axis (see
also Figures 3.13 and 3.14). Theorem 3.21 is formulated without the exact condition
for calculation for the number of stability switches, however these conditions (in form
ττ
τ
τ
τ
ττ
τ τ
τ
b
1
+
3
+
5
+
τ
τ
0
+
2
+
4
+
1
_
3
_
0
_
2
_
4
_
-sin( )2 ω
sin( )α απ
2 ω
α
ω
Figure 3.15: The stability region in
(b, τ)-plane for α = 1.8 and a = −1.
b
τ
Figure 3.16: The stability region in
(b, τ)-plane for α = 2 and a = −1.
Chapter 3. Analysis of two-term fractional delay differential equations 38
of a system of nonlinear inequalities) were given as a part of an example in [12].
(ii) Figure 3.15 depicts the stability region in the (b, τ)-plane showing clearly the
delay-independent stability region, the stability switching property as well as the
role of τ±
j (j ∈ Z+
0 ). Compare it to the Figures 3.8-3.10.
Let us consider α → 2−
and compare it to the known results for (1.6). The
asymptotes p+
α,m, p−
α,m tend to the lines b = −a − (mπ)2
/τ2
and b = a + (mπ)2
/τ2
,
which are the lines forming the stability boundary of (1.6) (see Theorem 1.9). Taking
α → 2−
in Theorem 3.21, we obtain the limit for the stability region in the form:
0 < |b| < −a and
ℓπ
−a − |b|
< τ <
(ℓ + 1)π
−a + |b|
where ℓ is a nonnegative integer that is even for b > 0 and odd for b < 0 (see
Figure 3.16). This form of conditions seems to be more effective compared to that
of Theorem 1.9, especially with respect to explicit evaluations of stability switches
for a varying delay parameter.
Chapter 4
Conclusions
In this thesis, we presented an in-depth exploration of the stability, asymptotic
behaviour, and oscillatory properties of several linear fractional differential problems
with a time delay. The focus was placed on optimal or near-optimal nature of
achieved results and on their explicit form in terms of system parameters whenever
possible. The introduced findings extend our understanding of nuances of fractional
and classical dynamics and in many cases they pioneered the qualitative theory of
fractional delay differential problems as outlined below.
We derived optimal stability conditions for the one-term FDDS of an arbitrary
order (2.1), including a comprehensive analysis at the stability boundary (see Theorems
2.9 and 2.10). Rather unconventionally, we also dealt with detailed asymptotic
description of unbounded solutions based on the location of system eigenvalues (see
Theorems 2.13 and 2.14). While stability properties display a smooth transition
across derivative orders, the asymptotic behaviour shows a striking contrast, as
can be expected based on theory of undelayed fractional differential equations. In
asymptotically stable cases, fractional derivatives lead to algebraic decay rates, as
opposed to the exponential decay seen in classical systems. As a consequence, solutions
to FDDS tend toward dominantly non-oscillatory behaviour which is a clear
difference from their integer-order counterparts (see Theorem 2.12 and 2.16).
For the two-term FDDE (3.1) of orders less than two, we described the stability
regions and grasped their evolution as the derivative order increases, passing through
classical integer cases (see Theorems 3.4, 3.19). Moreover, we provided several
insights into the emergence mechanism of stability switching phenomenon. Our
particular focus was on providing stability criteria in practical form, i.e. in terms
of entry coefficients, often in non-improvable versions (see Theorems 3.6, 3.14 and
3.21).
These theoretical results naturally transfer to practical applications, particularly
to control theory. They outline effective design strategies for stabilization or
destabilization of fractional systems via delayed feedback loops, as well as more nuanced
prediction of large-time behaviour or such system (see [11,12]). In the future,
we can expect emergence of other applications, e.g. in theory of complex systems
where nonlocal and memory-based nature of fractional derivatives in combination
– 39 –
Chapter 4. Conclusions 40
with time lagging seems to be a promising direction.
Regarding future research, there are several promising directions extending stability
and oscillatory analyses to more general cases. In particular, there are significant
opportunities in areas where the author already has substantial experience
in the undelayed context, such as discrete settings (see [7, 14, 34]), time-scale calculus
(see [18,33,36]), nonlinear dynamics and variable coefficients (see [8,35]), or
problems including multiple fractional operators (see [9]).
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Appendices
– 45 –
Appendix A
Paper on lower-order one-term
FDDS [6] (CNSNS, 2016)
Until 2016, I had already published nearly a dozen papers on fractional differential
and difference equations. However, it was the paper [6] (co-authors: J. Čermák, J.
Horníček; my author’s share 45 %) that expanded my scope to include equations
involving time delay. To this day, it remains my most cited work across all databases.
We focused on basics which were not sufficiently covered at the time. The stability
and asymptotic properties of autonomous linear FDDS of order less than one.
The main result was the formulation of necessary and sufficient conditions for stability
via the location of system matrix eigenvalues in complex plane. We also derived
algebraic decay rate of solutions tending to zero.
This paper laid the groundwork for techniques that we later employed for more
advanced and technically challenging problems. Notably, we have re-established
the fundamental solution for FDDS and introduced a generalized delay exponential
function of Mittag-Leffler type. Most importantly, we adopted a technique utilizing
the inverse Laplace transform and root analysis of the characteristic equation to
derive asymptotic behaviour of solutions. This approach proved crucial for our
subsequent research, as the presence of fractional derivatives disallows the direct use
of the link between the real part of characteristic roots and argument of exponentials
(since they do not belong among the solutions of fractional problems).
– 47 –
Commun Nonlinear Sci Numer Simulat 31 (2016) 108–123
Contents lists available at ScienceDirect
Commun Nonlinear Sci Numer Simulat
journal homepage: www.elsevier.com/locate/cnsns
Stability regions for fractional differential systems with a time
delay
Jan ˇCermák∗
, Jan Horníˇcek, Tomáš Kisela
Institute of Mathematics, Brno University of Technology, Technická 2, Brno CZ-616 69, Czech Republic
a r t i c l e i n f o
Article history:
Received 4 March 2015
Revised 21 July 2015
Accepted 22 July 2015
Available online 29 July 2015
Keywords:
Fractional delay differential equation
Stability
Asymptotic behaviour
a b s t r a c t
The paper investigates stability and asymptotic properties of autonomous fractional differential
systems with a time delay. As the main result, necessary and sufficient stability conditions
are formulated via eigenvalues of the system matrix and their location in a specific area of
the complex plane. These conditions represent a direct extension of Matignon’s stability criterion
for fractional differential systems with respect to the inclusion of a delay. For planar
systems, our stability conditions can be expressed quite explicitly in terms of entry parameters.
Applicability of these results is illustrated via stability investigations of the fractional
delay Duffing’s equation.
© 2015 Elsevier B.V. All rights reserved.
1. Introduction
Many problems described via ordinary differential equations require the inclusion of time delay terms. There are various types
of these delays (technological, transport, incubation and others) appearing in many science areas. Delays are often involved in
chemical processes (behaviours in chemical kinetics), technical processes (electric, pneumatic and hydraulic networks), biosciences
(heredity in population dynamics), economics (dynamics of business cycles) and other branches. In general, the distinguishing
feature of corresponding mathematical models is that the evolution rate of these processes depends on the past history.
The differential equations modelling these problems are called delay differential equations (DDEs). Basic qualitative theory of
these equations is well-established, especially in the linear case (for general references see [1] and [2]).
In the last decades, mathematical descriptions via fractional differential equations (FDEs) involving derivatives of non-integer
orders turned out to be a very useful tool in the modelling of various phenomena of viscoelasticity, anomalous diffusion, control
theory and other areas. Because of more degrees of freedom, fractional-order models are highly successful in situations where
non-Gaussian and non-Markovian processes occur. A survey of interesting applications as well as basic qualitative properties of
FDEs can be found, e.g. in the monographs [3–5].
Both these types of differential equations have a similar historical background. Although basic notions and properties related
to these equations were discussed already by the founders of classical calculus, a systematic study of these equations was missing.
It was only a rich application potential which gave rise to an enormous interest of mathematicians, engineers and other scientists
in the qualitative and numerical investigations of DDEs and FDEs (starting in sixties of the last century).
The unification of DDEs and FDEs is provided by fractional delay differential equations (FDDEs), involving both the delay
and non-integer derivative terms and disposing of great complexity. In technical applications, this approach is useful in creating
strongly realistic models of certain processes and systems with memory and heredity. It is involved, among others, in analysis
∗
Corresponding author. Tel.: +420 5411 42535.
E-mail addresses: cermak.j@fme.vutbr.cz (J. ˇCermák), y115594@stud.fme.vutbr.cz (J. Horníˇcek), kisela@fme.vutbr.cz (T. Kisela).
http://dx.doi.org/10.1016/j.cnsns.2015.07.008
1007-5704/© 2015 Elsevier B.V. All rights reserved.
J. ˇCermák et al. / Commun Nonlinear Sci Numer Simulat 31 (2016) 108–123 109
of various time delay systems whose stabilisation and control is realised via a state feedback. Following the recent trend, these
controllers are often proposed using the fractional-order integro-differentiation method which gives rise to various types of
FDDEs (e.g., for a fractional delay state space model of PDα control of Newcastle robot we refer to [6]). This originally purely
theoretical idea of fractional-order controllers was recently supported and justified by several papers studying experimentally
collected impedance data of supercapacitors whose underlying electrochemical dynamics can be described and captured using
of fractional-order models (see, e.g. [7] and the papers cited therein).
The stability issue is standardly of the main interest in the control theory and other areas where DDEs and FDEs play a significant
role. The presence of time delay terms in feedback control systems results in characteristic equations involving transcendental
terms of the exponential type. The corresponding stability polynomials (usually called quasi-polynomials) have infinitely
many isolated zeros and analysis of their location (necessary for stability analysis of time invariant delay systems) is often a complicated
matter. If these systems involve non-integer derivative terms, then the situation becomes even more difficult. As noted
by several authors (see, e.g. [8]), the existing stability conditions for FDDEs do not provide effective algebraic criteria or algorithms
for testing of stability of FDDEs and they are difficult to use in practice (see, e.g. [9]). From this viewpoint, the lack of such
algebraic algorithms has hindered the advance of FDDEs for designs of control systems. Therefore, the main goal of this paper is
to fill in this gap and present a simple and easily verifiable criterion for stability testing of linear time invariant fractional delay
systems, including its rigorous mathematical justification (which seems to be extendable also to more general types of FDDEs).
The paper is organised as follows. In Section 2, we formulate our main result, namely effective stability criterion for a linear
system of FDDEs. A brief survey of existing results on this topic is included as well. Section 3 shortly recapitulates a necessary
background, mainly concerning fractional calculus and the Laplace transform. Section 4 discusses some properties of the studied
system of FDDEs which are important in our investigations. In Section 5, we consider the characteristic equation for this system
and analyse locations of its zeros in the complex plane. Using previous auxiliary assertions, Section 6 completes the proof of our
main stability criterion. In Section 7, we reformulate this criterion for the corresponding planar system in terms of trace and
determinant of the system matrix. Also, we apply the obtained results to stability investigations of the fractional delay Duffing’s
model of a nonlinear oscillator. Some remarks concerning future perspectives conclude the paper.
2. The main result
In the linear case, the fractional delay system
Dα
0 y(t) = Ay(t − τ), t ∈ (0, ∞), (2.1)
where Dα
0 is the Caputo derivative of a real order 0 < α < 1, A ∈ Rd×d is a constant real d × d matrix, y : ( − τ, ∞) → Rd and τ >
0 is a constant real lag, may serve as the basic prototype of FDDEs. The standard initial condition associated with (2.1) is
y(t) = φ(t), t ∈ [−τ, 0] (2.2)
where φ(t) ∈ L1([−τ, 0]), i.e. all components of φ(t) are absolutely Riemann integrable on [−τ, 0].
Contrary to DDEs and FDEs, a general qualitative theory of FDDEs is just at the beginning. During the past years, several
pioneering works on this topic appeared, with an emphasis put on stability issues. We recall that the zero solution of (2.1) is said
to be stable (asymptotically stable) if for any φ(t) ∈ L1([−τ, 0]) the solution y(t) of (2.1), (2.2) is bounded (tends to zero as t →
∞). Sufficient asymptotic stability conditions for (2.1) with multiple fractional orders and multiple delays have been formulated
in [9] and [10] via the zeros location of associated characteristic equations. In the scalar case, asymptotic stability properties of
Dα
0 y(t) = λy(t − τ), t ∈ (0, ∞), (2.3)
where λ is a real number, have been discussed in [11] by use of a transcendent inequality involving the fractional Lambert
function. The bounded input–bounded output (BIBO) stability regions for (2.3), involving also the non-delayed term y(t) on
its right-hand side, have been investigated in [12] where the D-decomposition method was employed to describe the stability
boundary of the studied equation in the form of parametric equations.
The first explicit stability criterion for (2.3) appeared in Theorem 5.1 of [13]. We recall here the relevant result in the following
assertion (the symbol ∼ used here means an asymptotic equivalence in the sense that the ratio of both involved terms tends to a
nonzero finite constant).
Theorem 1. Let λ ∈ R, 0 < α < 1 and τ > 0.
(i) The zero solution of (2.3) is asymptotically stable if and only if
−
π − απ/2
τ
α
< λ < 0. (2.4)
Moreover, y(t) ∼ t−α as t → ∞ for any solution y(t) of (2.3).
(ii) The zero solution of (2.3) is stable if and only if
−
π − απ/2
τ
α
≤ λ ≤ 0. (2.5)
Note that for α = 1, the condition (2.4) becomes −π/2 < λτ < 0 which is the classical asymptotic stability criterion for the
first order delay equation y (t) = λy(t − τ).
110 J. ˇCermák et al. / Commun Nonlinear Sci Numer Simulat 31 (2016) 108–123
Fig. 1. The stability region Sα,τ for the values α = 0.4 and τ = 1 (including tangents of the stability boundary at the origin).
The main goal of this paper is to follow the previous works and, in particular, to provide a vector extension of Theorem 1.
Doing this, we introduce the stability set
Sα,τ = λ ∈ C : |λ| <
|arg (λ)| − απ/2
τ
α
, |arg (λ)| >
απ
2
where we assume −π < arg ( · ) ≤ π. Throughout this paper, we utilise the usual notation for its closure cl(Sα,τ ) and its boundary
∂Sα,τ . Then we have the following generalisation of Theorem 1 (the symbol · means a norm in Rd).
Theorem 2. Let A ∈ Rd×d, 0 < α < 1 and τ > 0.
(i) The zero solution of (2.1) is asymptotically stable if and only if all eigenvalues λ of A are located inside Sα,τ . Moreover, y(t) ∼ t−α
as t → ∞ for any solution y(t) of (2.1).
(ii) The zero solution of (2.1) is stable if and only if all eigenvalues λ of A belong to cl(Sα,τ ) and all eigenvalues lying on ∂Sα,τ have the
same algebraic and geometric multiplicities.
Remark 1. (a) Theorem 2 immediately implies that if A = λ is a real scalar, then the corresponding stability and asymptotic
stability conditions become (2.5) and (2.4), respectively. In this connection, we consider two other important limit cases of (2.1),
namely for α → 1− and τ → 0+ when (2.1) becomes the first order linear DDE
y (t) = Ay(t − τ), t ∈ (0, ∞) (2.6)
and the α-order linear FDE
Dα
0 y(t) = Ay(t), t ∈ (0, ∞), (2.7)
respectively. If we set α = 1 in Theorem 2, then we get just stability conditions formulated in Theorem 3.4 of [14] for the delay
system (2.6). Similarly, if τ → 0+, then the conditions of Theorem 2 are reduced to the known Matignon’s stability condition
|arg (λ)| > απ/2 which is taken for the starting point in stability analysis of FDEs (see [15]).
(b) The stability regions Sα,τ for (2.1) and their dependance on parameters α, τ are depicted in Figs. 1–4. In particular, we
can observe here that Sα,τ approaches the unit circle |λ| < 1 as α → 0+ which is the well-known stability area for the discrete
system yn = Ayn−τ , n = τ, 2τ, . . . (see Fig. 4). From this stability viewpoint, (2.1) provides a bridge between the delay differential
system (2.6) and this discrete system.
3. Some preliminaries
Let f(t) be a real function. Throughout this paper, we employ the standard definitions of the fractional integral
D
−γ
a f(t) =
t
a
(t − ξ)γ −1
(γ )
f(ξ)dξ, γ > 0, a ∈ R, t ≥ a
and the Caputo fractional derivative
Dα
a f(t) = D−(1−α)
a
d
dt
f(t) , 0 < α < 1, a ∈ R, t ≥ a
where we put D0
a f(t) = f(t) (for more information on fractional operators we refer, e.g. to [3,5]).
J. ˇCermák et al. / Commun Nonlinear Sci Numer Simulat 31 (2016) 108–123 111
Fig. 2. The stability region Sα,τ for the values α = 0.4 and τ = 0.1 (the corresponding limit case as τ → 0+
is dotted).
Fig. 3. The stability region Sα,τ for the values α = 0.9 and τ = 1 (the corresponding limit case as α → 1−
is dotted).
In this connection, we note that the notion of FDDEs opens a space for discussions concerning the proper choice of lower limit
of the fractional derivative. In particular, it is worth to consider the possibility of putting this limit a = −τ. This issue is closely
related to the problem of initialisation introduced and developed by Lorenzo and Hartley (see, e.g. [16]). Although a deeper
analysis of this issue extends the scope of this paper, we note that such an adjustment can be viewed as adding a forcing term on
the right-hand side of (2.1), which does not affect stability and asymptotic behaviour of the solution due to its decay rate t−α. In
this paper, we adopt the standard approach and consider the lower limit of fractional operators to be zero, i.e. a = 0.
In our next considerations, we utilise the well-known Laplace transform of f(t) introduced as
L(f(t))(s) =
∞
0
exp{−st}f(t)dt, s ∈ C
provided the integral converges. For the sake of lucidity, we recall some relevant basic formulae, namely
L(f(t − τ)h(t − τ))(s) = exp{−sτ}L(f(t))(s), τ > 0, (3.1)
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Fig. 4. The stability region Sα,τ for the values α = 0.05 and τ = 1 (the corresponding limit case as α → 0+
is dotted).
L(f(t − τ))(s) = exp{−sτ}L(f(t))(s) + exp{−sτ} 0
−τ exp{−su}f(u)du, τ > 0 (3.2)
where h( · ) is the Heaviside step function defined as h(ξ) = 1 for ξ ≥ 0 and h(ξ) = 0 for ξ < 0. Furthermore, we also recall the
well-known convolution formula for real functions f(t), g(t)
L
t
0 f(t − ξ)g(ξ)dξ (s) = L(f(t))(s) · L(g(t))(s), (3.3)
which is useful, among others, in procedures related to the Laplace inverse L−1.
In the frame of fractional calculus, a key relationship is provided by the Laplace transform of a power function
L
tη
(η + 1)
(s) = s−η−1
, η > −1. (3.4)
This along with (3.3) leads to the Laplace transforms of fractional operators in the form
L(D
−γ
0
f(t))(s) = s−γ L(f(t))(s), γ > 0, (3.5)
L(Dα
0 f(t))(s) = sαL(f(t))(s) − sα−1
f(0), 0 < α < 1 (3.6)
(see, e.g. [5]). Note also that the Laplace transform of vector and matrix functions is considered componentwise.
In stability analysis of (2.1), it is convenient to employ the Jordan canonical form of the system matrix A. On this account, we
recall that every d × d matrix A is similar to a d × d matrix with the Jordan blocks on its diagonal, i.e.
A = T T−1
, =
⎛
⎜
⎜
⎝
J1 0 · · · 0
0 J2 · · · 0
...
...
...
...
0 0 · · · Jq
⎞
⎟
⎟
⎠, Jk =
⎛
⎜
⎜
⎜
⎜
⎜
⎝
λi 1 0 · · · 0
0 λi 1
...
...
...
...
...
... 0
0 · · · λi 1
0 · · · 0 λi
⎞
⎟
⎟
⎟
⎟
⎟
⎠
, k = 1, . . . , q (3.7)
where T is an invertible matrix and λi (i = 1, . . . , n) are distinct eigenvalues of A. The number of Jordan blocks corresponding to
λi is called the geometric multiplicity of λi. The sum of the sizes of all Jordan blocks corresponding to λi is called the algebraic
multiplicity of λi.
Besides the symbol ∼ for the asymptotic equivalence of given functions (recalled in the introductory part), we employ also
another asymptotic notation
f ∼sup g as t → ∞ ⇐⇒ lim sup
t→∞
f(t)
g(t)
= K = 0
which enables to describe asymptotics of a wider class of functions (in particular, unbounded oscillatory functions).
J. ˇCermák et al. / Commun Nonlinear Sci Numer Simulat 31 (2016) 108–123 113
4. Solution representation for (2.1) and its properties
In this section, we discuss a suitable representation of the solution of (2.1), (2.2). Similarly to the integer-order case (see, e.g.
[1,2]), the essential role in this matter is played by the so-called fundamental matrix solution.
Definition 1. Let A ∈ Rd×d and let I be the identity d × d matrix. The matrix function R : R → Cd×d given by
R(t) = L−1
(sαI − A exp{−sτ})−1
(t)
is called the fundamental matrix solution of (2.1).
Remark 2. In the integer-order case, the fundamental matrix solution is usually considered to be a matrix function whose ith
column solves the correspoding delay system supplied with the initial condition φ(t) = 0 for t ∈ [−τ, 0) and φ(0) = ei, where ei
is the standard basis vector in Rd. Such an introduction of the fundamental matrix solution is possible also in the fractional-order
case (see, e.g. [13]). In our investigations, it does not seem to be convenient especially due to some difficulties connected with the
variation of constants formula. On this account, we prefer Definition 1 which enables us to simplify the main parts of the proofs.
We stress that Definition 1 for α → 1− coincides with the standard integer-order definition of the fundamental matrix solution.
The notion of fundamental matrix solution leads to the following solution representation.
Theorem 3. The solution y(t) of (2.1), (2.2) is given by
y(t) = D−(1−α)
0
R(t)φ(0) +
0
−τ
R(t − τ − u)h(t − τ − u)Aφ(u)du , t > 0 , (4.1)
where h(ξ) is the Heaviside step function.
Proof. Applying (3.2) and (3.6), we easily obtain the Laplace image of the solution of (2.1), (2.2) in the form
L(y(t))(s) = (sαI − A exp{−sτ})−1
sα−1
φ(0) + exp{−sτ}
0
−τ
exp{−su}Aφ(u)du
= sα−1
L(R(t))(s)φ(0) +
0
−τ
exp{−s(τ + u)}L(R(t))(s)Aφ(u)du.
Then, using (3.5) for the first term and employing linearity of the Laplace transform and (3.1) with respect to the expression
exp{−s(τ + u)}L(R(t))(s) for the second term, we get the assertion.
Remark 3. We point out that (4.1) is equivalent to the expressions obtained in [12] and [13] using a different definition of the
fundamental matrix solution.
It is well-known that the solution of (2.7) can be expressed via functions of Mittag–Leffler type (see, e.g. [5]). Following this,
we introduce
Definition 2. Let λ ∈ C, η, β, τ ∈ R and m ∈ Z be such that η, β, τ > 0, m ≥ 0. The generalised delay exponential function (of
Mittag–Leffler type) is given by
Gλ,τ,m
η,β (t) =
∞
j=0
m + j
j
λj
(t − (m + j)τ)η(m+ j)+β−1
(η(m + j) + β)
h(t − (m + j)τ), t > 0 (4.2)
where h(ξ) is the Heaviside step function.
Remark 4. We note that the sum (4.2) is infinite only formally. In fact, for every fixed t > 0, all terms satisfying m + j > t/τ are
equal to zero due to the occurrence of the Heaviside step function.
Lemma 1. Let λ ∈ C, η, β, τ ∈ R and m ∈ Z be such that η, β, τ > 0, m ≥ 0. Then it holds
L(Gλ,τ,m
η,β (t))(s) = sη−β exp{−msτ}
(sη−λexp{−sτ})m+1 .
Proof. Using linearity of the Laplace transform and the formulae (3.1) and (3.4), we get
L(Gλ,τ,m
η,β (t))(s) =
∞
j=0
λj m + j
j
exp{−(m + j)sτ}
1
sη(m+ j)+β
= exp{−mτs}s−mη−β
∞
j=0
m + j
j
λ
sη exp{sτ}
j
.
Then we can employ the property
m + j
j
= ( − 1)j −m − 1
j
, which is valid for all nonegative integers m, j, to obtain
L(Gλ,τ,m
η,β (t))(s) = exp{−mτs}s−mη−β
∞
j=0
( − 1)j −m − 1
j
λ
sη exp{sτ}
j
.
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Furthermore, application of the binomial formula yields
L(Gλ,τ,m
η,β (t))(s) =
exp{−mτs}s−mη−β
1 − λ
sη exp{sτ}
m+1
=
sη−β exp{−msτ}
(sη − λexp{−sτ})m+1
,
which concludes the proof.
Now, we are in a position to formulate the main result of this section.
Theorem 4. Let R(t) be the fundamental matrix solution of (2.1). Furthermore, let λi (i = 1, . . . , n) be distinct eigenvalues of A and let
pi be the largest dimension of the Jordan block corresponding to the eigenvalue λi. Then the nonzero elements of R(t) are given by linear
combinations of the generalised delay exponential functions
Gλi,τ,m
α,α (t), m = 0, . . . , pi − 1, i = 1, . . . , n.
Proof. Using (3.7) we can rewrite the fundamental matrix solution of (2.1) as
R(t) = L−1
(sαI − A exp{−sτ})−1
(t) = T−1
L−1
(sαI − exp{−sτ})−1
(t)T (4.3)
where T is a constant invertible matrix and is the Jordan canonical form of A. Thus, R(t) is formed by linear combinations
of elements of the fundamental matrix solution of (2.1) with A replaced by . Since is the block diagonal matrix, (sαI −
exp{−sτ})−1 is block diagonal as well. Moreover, its blocks are upper triangular strip matrices of the type
(sαI − Jke−sτ )−1
=
⎛
⎜
⎜
⎜
⎝
(sα − λie−sτ )−1
e−sτ (sα − λie−sτ )−2
· · · e−(rk−1)sτ (sα − λie−sτ )−rk
0 (sα − λie−sτ )−1
... e−(rk−2)sτ (sα − λie−sτ )−(rk−1)
...
...
...
...
0 0 · · · (sα − λie−sτ )−1
⎞
⎟
⎟
⎟
⎠
(4.4)
where Jk (k = 1, . . . , q) is the kth block of and rk is its dimension. Lemma 1 implies that all involved elements are Laplace
transforms of the functions stated in the assertion.
5. The characteristic equation and its analysis
It is known from theories of DDEs and FDEs that positions of singular points of the Laplace image of the solutions indicate
stability properties (and in some cases also a decay rate of the solutions) of the original equation. In Section 6, we show that a
similar connection occurs in the case of FDDEs. As a preliminary step, this section is devoted to the pole analysis of the Laplace
image of the fundamental matrix solution of (2.1). In other words, we discuss locations of zeros of
det (sαI − A exp{−sτ}) = 0 (5.1)
which is called the characteristic equation of (2.1). We can see from (4.3) and (4.4) that, instead of (5.1), it is sufficient to investigate
the zeros of the equation
p(s) ≡ sα − λexp{−sτ} = 0 (5.2)
where λ is a complex parameter.
The standard way to prove stability criteria for linear autonomous DDEs is based on the D-decomposition method applied to
the corresponding characteristic quasi-polynomials. Although this procedure is applicable also in the fractional-order case, we
prefer an alternative method based on a direct zero analysis of (5.2).
The basic zero properties of (5.2) can be collected in the following
Proposition 1.
(i) The complex number s is a zero of (5.2) if and only if its complex conjugate s∗ is a zero of
sα − λ∗
exp{−sτ} = 0
where λ∗ is complex conjugate to λ.
(ii) There exists δ > 0 such that (5.2) has only a finite number of zeros s with |arg(s)| < π/2 + δ.
(iii) All zeros of (5.2) have negative real parts if and only if λ ∈ Sα,τ .
(iv) If s1, s2 are zeros of (5.2) satisfying (s1) = (s2), then |s1| = |s2|.
(v) Let s be a zero of (5.2), i.e. p(s) = 0. Then p (s) = 0 for all s with (s) = 0 or (s) > 0.
Proof. We assume λ = 0 (the opposite case is trivial). Let s = r exp{iϕ}, λ = exp{iψ} for r = |s|, = |λ| and suitable −π < ϕ ≤
π, −π < ψ ≤ π. Then (5.2) takes the form
rα exp{iαϕ} − exp{iψ}exp{−rτ exp{iϕ}} = 0.
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We equate the real and imaginary parts to obtain
rα cos (αϕ) − exp{−rτ cos (ϕ)}cos (ψ − rτ sin (ϕ)) = 0, (5.3)
rα sin (αϕ) − exp{−rτ cos (ϕ)}sin (ψ − rτ sin (ϕ)) = 0. (5.4)
If ϕ = 0, then 0 < α|ϕ| < π, hence ψ − rτ sin (ϕ) = kπ for any k ∈ Z. Dividing (5.3) over (5.4) we obtain
cotan(αϕ) = cotan(ψ − rτ sin (ϕ)), i.e. αϕ = ψ − rτ sin (ϕ) + kπ, k ∈ Z. (5.5)
If we substitute (5.5) into (5.4), we get
rα = ( − 1)k
exp{−rτ cos (ϕ)}, k ∈ Z
where k must be even due to positivity of r and ϱ.
If ϕ = 0, then ψ = 0 due to (5.4) and (5.3), i.e. λ = and (5.2) is reduced to
rα = exp{−rτ}.
Consequently, summarising this part, s is a zero of (5.2) if and only if its modulus r and argument ϕ solve the system
αϕ = ψ − rτ sin (ϕ) + 2kπ, (5.6)
rα = exp{−rτ cos (ϕ)} (5.7)
for a suitable k ∈ Z. This reformulation turns out to be useful in our analysis of (i)–(iv). While the properties (i) and (iv) follow
from it directly, the proof of (ii) and (iii) requires some additional steps.
If 0 < |ϕ| < π, then (5.6) yields
r =
ψ − αϕ + 2kπ
τ sin (ϕ)
> 0 (5.8)
and, substituting (5.8) into (5.7), we get
ψ − αϕ + 2kπ
τ sin (ϕ)
α
= exp{( − ψ + αϕ − 2kπ)cotan(ϕ)}.
To analyse this equality, we set
gk(ϕ) =
ψ − αϕ + 2kπ
τ sin (ϕ)
α
, hk(ϕ) = exp{( − ψ + αϕ − 2kπ)cotan(ϕ)}
and discuss intersections of gk(ϕ) and hk(ϕ) in (0, π/2]. Obviously, the function gk(ϕ) is decreasing in (0, π/2] with
lim
ϕ→0+
gk(ϕ) = ∞ and gk(π/2) =
ψ − απ/2 + 2kπ
τ
α
.
Similarly, hk(ϕ) is increasing in (0, π/2]. Indeed, using (5.8) we get
d
dϕ [( − ψ + αϕ − 2kπ)cotan(ϕ)] = α cotan(ϕ) + ψ−αϕ+2kπ
sin2
(ϕ)
= α cotan(ϕ) + rτ
sin (ϕ) > 0
in (0, π/2]. Moreover,
lim
ϕ→0+
hk(ϕ) = 0 and hk(π/2) = .
Consequently, if m1 is the smallest non-negative integer such that
<
ψ − απ/2 + 2m1π
τ
α
,
then, because of the continuity of gk(ϕ) and hk(ϕ) in (0, π), the system (5.6) and (5.7) has just m1 solutions (r, ϕ) with ϕ ∈
(0, π/2 + δ), δ > 0 being sufficiently small. Equivalently, (5.2) has m1 zeros s with 0 < arg(s) < π/2 + δ. Analogously, (5.2) has
m2 zeros s with −(π/2 + δ) < arg(s) < 0 where m2 is the smallest non-negative integer such that
<
−ψ − απ/2 + 2m2π
τ
α
.
Furthermore, as noted above, (5.2) has a (unique) real zero if and only if λ is a positive real, i.e. λ = . This proves (ii).
To complete the proof of (iii), it is enough to repeat the previous argumentation. In particular, the functions gk(ϕ) and hk(ϕ)
have no intersections in (0, π/2] for any k = 0, 1, . . . if and only if gk(π/2) > hk(π/2), i.e.
<
ψ − απ/2 + 2kπ
τ
α
116 J. ˇCermák et al. / Commun Nonlinear Sci Numer Simulat 31 (2016) 108–123
for any k = 0, 1, . . .. Taking into account positivity of r in (5.8), this inequality holds if and only if
<
ψ − απ/2
τ
α
and ψ >
απ
2
.
Using = |λ|, ψ = arg(λ) and the property (i), these conditions imply λ ∈ Sα, τ .
Finally, the proof of (v) follows from a direct calculation. Indeed, the relations p(s) = p (s) = 0 imply
sα − λexp{−sτ} = αsα−1
+ λτ exp{−sτ} = 0
which can be considered as the system of two equations with unknown s and λ. Its simple analysis yields (v).
Remark 5. The property (i) implies, among others, the symmetry of the area Sα,τ with respect to the real axis. More precisely,
the asymptotic stability boundary, when two zeros of (5.2) are purely imaginary and all the remaining zeros have negative real
parts, is the curve symmetric with respect to the line (λ) = 0 and given via the parametric equations in the complex λ-plane
(λ) = |ω|α cos |ω|τ +
απ
2
, (λ) = sgn(ω)|ω|α sin |ω|τ +
απ
2
, −
π − απ/2
τ
≤ ω ≤
π − απ/2
τ
.
6. Proof of Theorem 2
We start this section with the following auxiliary, but essential result describing the asymptotic behaviour of the generalised
delay exponential functions.
Lemma 2. Let λ ∈ C, α, β, τ ∈ R and m ∈ Z be such that 0 < α < 1, β, τ > 0, m ≥ 0. Furthermore, let si (i = 1, 2, . . .) be the zeros of
(5.2) with ordering (si) ≥ (si+1) (in particular, s1 is the zero with the largest real part).
(i) If λ = 0, then
G0,τ,m
α,β (t) =
(t − mτ)mα+β−1
(mα + β)
h(t − mτ).
(ii) If λ ∈ Sα,τ , then
Gλ,τ,m
α,β (t) =
( − 1)m+1
λm+1 (β − α)
(t + τ)β−α−1
+
( − 1)m+1
(m + 1)
λm+2 (β − 2α)
(t + 2τ)β−2α−1
+ O(tβ−3α−1
) as t → ∞.
(iii) If λ /∈ Sα,τ ∪ {0}, then
Gλ,τ,m
α,β (t) =
m
j=0
(t − mτ)j
(aj exp{s1(t − mτ)} + bj exp{s2(t − mτ)})
+
O(tm
exp{ (s3)t}), if (s3) ≥ 0,
O(tβ−α−1
), if (s3) < 0
as t → ∞
where aj, bj are suitable nonzero complex constants (j = 0, . . . , m).
Proof. If λ = 0, then Definition 2 implies directly the assertion (i). Let λ = 0. By Lemma 1, it holds
L(Gλ,τ,m
α,β (t))(s) =
sα−β exp{−msτ}
(sα − λexp{−sτ})m+1
.
We can see that L(Gλ,τ,m
α,β (t))(s) has poles si (i = 1, 2, . . .) corresponding to zeros of (5.2) and, if β > α, then there is also a pole
at s0 = 0. Furthermore, the occurrence of the power function implies that the negative real axis consists of singular points only
(zero is a branch point). The inverse Laplace transform formula yields
Gλ,τ,m
α,β (t) =
1
2πi
c+i∞
c−i∞
exp{st}
sα−β exp{−msτ}
(sα − λexp{−sτ})m+1
ds,
where c ∈ R satisfies c > max (0, (s1)), i.e. all singularities of the previous integrand are located to the left of the line (s) = c.
Obviously, instead of the line (s) = c, we can use an arbitrary curve with the same property (we note that the idea of some
next proof procedures originates from the method presented in [5] for the asymptotic description of Mittag–Leffler functions).
In particular, we are going to employ oriented contours formed by three segments given via
γ (ζ, θ) = s ∈ C : s = −u exp{−iθ}, u ∈ ( − ∞, −ζ) or s = ζ exp{−iu}, u ∈ [−θ, θ] or s = u exp{−iθ}, u ∈ (ζ, ∞) .
Proposition 1 (ii) implies that there exists δ > 0 such that all zeros si of (5.2) satisfy |arg (si)| = π/2 + δ and, moreover, that
there are only finitely many of them satisfying |arg (si)| < π/2 + δ. Hence, there exist R > ε > 0 such that all si lie to the left of
γ (R, π/2 + δ) and those satisfying |arg (si)| < π/2 + δ are located to the right of γ (ε, π/2 + δ). Consequently, for every t > 1,
we can split the inverse Laplace transform formula into
Gλ,τ,m
α,β (t) =
1
2πi γ (R, π
2 +δ)
sα−β exp{st − msτ}
(sα − λexp{−sτ})m+1
ds = I1(t) + I2(t)
where I1(t) and I2(t) denote the integrals over γ (ε/t, π/2 + δ) and γ (R, π/2 + δ) − γ (ε/t, π/2 + δ), respectively.
J. ˇCermák et al. / Commun Nonlinear Sci Numer Simulat 31 (2016) 108–123 117
First, we analyse the integral I1(t). Employing the change of variables s = u1/α/t, which transforms the contour σ =
γ (ε/t, π/2 + δ) into σ = γ (εα, απ/2 + αδ), and the identity
1
ξ − z
m+1
= −
1
z
−
ξ
z2
+
ξ2
z2(ξ − z)
m+1
=
( − 1)m+1
zm+1
+ (m + 1)
( − 1)m+1
ξ
zm+2
+
(k1,k2,k3)∈K
(m + 1)!
k1!k2!k3!
( − 1)k1+k2 ξk2+2k3
zk2+k3+m+1(ξ − z)k3
,
where K = {(k1, k2, k3) ∈ (Z+
0
)3 : k1 + k2 + k3 = m + 1, (k1, k2, k3) = (m + 1, 0, 0) and (k1, k2, k3) = (m, 1, 0)} (for the multinomial
theorem see, e.g. [17]), we obtain
I1(t) =
1
2πi σ
sα−β exp{st − msτ}
(sα − λexp{−sτ})m+1
ds =
tmα+β−1
2παi σ
u(1−β)/α exp{(1 + τ/t)u1/α}
(u exp{u1/ατ/t} − λtα)m+1
du
=
tmα+β−1
2παi
( − 1)m+1
σ
u(1−β)/α exp{(1+τ/t)u1/α}
(λtα)m+1
du+( − 1)m+1
(m+1)
σ
u(1−β)/α+1
exp{(1+2τ/t)u1/α}
(λtα)m+2
du
+
(k1,k2,k3)∈K
(m + 1)!
k1!k2!k3! σ
( − 1)k1+k2
(λtα)k2+k3+m+1
u(1−β)/α+k2+2k3 exp{(1 + (1 + k2 + 2k3)τ/t)u1/α}
(u exp{u1/ατ/t} − λtα)k3
du .
Furthermore, we use the change of variables v = (1 + kτ/t)αu, transforming σ into σk
= γ ((1 + kτ/t)αεα, απ/2 + αδ) (k =
1, 2), and the repeated application of the integral representation of the reciprocal Euler -function
1
(z)
=
1
2παi γ (ζ,θ)
exp{ξ1/α}ξ1/α−z/α−1
dξ, ζ > 0, πα/2 < θ < απ
(see [5]) to get
I1(t) = ( − 1)m+1 (t + τ)β−α−1
2παλm+1i σ1
v(1−β)/α exp{v1/α}dv+( − 1)m+1
(m+1)
(t+2τ)β−2α−1
2παλm+2i σ2
v(1−β)/α+1
exp{v1/α}dv
+
(k1,k2,k3)∈K
tβ−(k2+k3+1)α−1
2παλk2+k3+m+1i
(m + 1)!
k1!k2!k3! σ
( − 1)k1+k2
u(1−β)/α+k2+2k3 exp{(1 + (1 + k2 + 2k3)τ/t)u1/α}
(u exp{u1/ατ/t} − λtα)k3
du
=
( − 1)m+1
λm+1 (β − α)
(t + τ)β−α−1
+
( − 1)m+1
(m + 1)
λm+2 (β − 2α)
(t + 2τ)β−2α−1
+ I3(t)
where I3(t) denotes the last term in the previous expression for I1(t). In order to discuss the asymptotic behaviour of I3(t), we first
formulate the following estimate. Since the contour σ does not contain any zero si of (5.2), there exist δk > 0 (k = 0, . . . , m + 1)
such that |sα − λexp{−sτ}|k ≥ δk for all s ∈ σ. Consequently, we have
|u exp{u1/ατ/t} − λtα|k
≥ δktkα|exp{u1/ατ/t}|k
for all u ∈ σ .
It enables us to write
|I3(t)| ≤
(k1,k2,k3)∈K
tβ−(k2+k3+1)α−1
2πα|λ|k2+k3+m+1
(m + 1)!
k1!k2!k3! σ
u(1−β)/α+k2+2k3 exp{(1 + (1 + k2 + 2k3)τ/t)u1/α}
(u exp{u1/ατ/t} − λtα)k3
du
≤
(k1,k2,k3)∈K
tβ−(k2+2k3+1)α−1
2πα|λ|k2+k3+m+1δk3
(m + 1)!
k1!k2!k3! σ
u(1−β)/α+k2+2k3
exp{(1 + (1 + k2 + k3)τ/t)u1/α} du
≤
(k1,k2,k3)∈K
tβ−(k2+2k3+1)α−1
2πα|λ|k2+k3+m+1δk3
(m + 1)!
k1!k2!k3!
απ/2+αδ
−απ/2−αδ
ε(1−β)+(1+k2+2k3)α
× exp{(1 + (1 + k2 + k3)τ/t)ε cos (ϕ/α)}dϕ
+ 2
∞
εα
r(1−β)/α+k2+2k3
exp{(1 + (1 + k2 + k3)τ/t)r1/α cos (π/2 + δ)}dr .
We can see that the last inequality contains real integrals of two types, namely the integrals of functions continuous on the
compact integration domain [−απ/2 − αδ, απ/2 + αδ] (which implies their convergence) and the integrals over the unbounded
domain (εα, ∞) which converge due to cos (π/2 + δ) < 0.
Now we turn our attention to the second term I2(t). Clearly, the contour γ (R, π/2 + δ) − γ (ε/t, π/2 + δ) is a simple positively
oriented closed curve. Proposition 1 (ii) implies that the integrand has only finitely many poles si (i = 1, . . . , N) lying in the interior
of γ (R, π/2 + δ) − γ (ε/t, π/2 + δ). Moreover, it follows from Proposition 1 (v) that si are poles of order m + 1. Employing the
118 J. ˇCermák et al. / Commun Nonlinear Sci Numer Simulat 31 (2016) 108–123
residue theorem, we get
I2(t) =
1
2πi γ (R,π/2+δ)−γ (ε/t,π/2+δ)
sα−β exp{st − msτ}
(sα − λexp{−sτ})m+1
ds =
N
i=1
Ress=si
sα−β exp{st − msτ}
(sα − λexp{−sτ})m+1
.
At every pole si we can utilise the Laurent expansions
sα−β
(sα − λexp{−sτ})m+1
= c−m−1(s − si)−m−1
+ c−m(s − si)−m
+ c−m+1(s − si)−m+1
+ . . . ,
exp{st − msτ} = exp{si(t − mτ)} 1 + (t − mτ)(s − si) +
(t − mτ)2
2!
(s − si)2
+
(t − mτ)3
3!
(s − si)3
+ . . .
where cj ( j = −m − 1, −m, . . .) are complex constants independent of t and c−m−1 = 0. Thus, the Cauchy product of these expansions
enables us to write the sum of residues (i.e. the coefficients at (s − si)−1) as
I2(t) = N
i=1
m
j=0 di,j(t − mτ)j
exp{si(t − mτ)}
where di, j are suitable complex constants and di, m = 0 (i = 1, . . . , N, j = 0, . . . , m).
Combining the obtained results, we get
Gλ,τ,m
α,β (t) =
( − 1)m+1
λm+1 (β − α)
(t + τ)β−α−1
+
( − 1)m+1
(m + 1)
λm+2 (β − 2α)
(t + 2τ)β−2α−1
+
N
i=1
m
j=0
di,j(t − mτ)j
exp{si(t − mτ)} + O(tβ−3α−1
)
as t → ∞. Proposition 1 (iii) implies that for λ ∈ Sα,τ the exponential terms vanish as t → ∞ and the part (ii) of the assertion is
proved. If λ /∈ Sα,τ , then the exponential terms play the dominant role in asymptotic investigations. Moreover, Proposition 1 (iv)
and (v) yields that there are at most two poles si of the maximal real part, i.e. it holds (s3) < (s1). This concludes the proof of
the part (iii).
Now we have all the partial results necessary to prove the main result.
Proof of Theorem 2. Theorems 3 and 4 imply that every coordinate of the solution y(t) of (2.1), (2.2) consists of a linear combination
of terms
D−(1−α)
0
Gλi,τ,m
α,α (t),
0
−τ
Gλi,τ,m
α,α (t − τ − u)φ(u)du, m = 0, . . . , pi − 1, i = 1, . . . , n
where λi are distinct eigenvalues of A and pi are the maximal sizes of Jordan blocks corresponding to λi.
We analyse the stability properties of these terms. First, we consider the (1 − α)-integral of G
λi,τ,m
α,α (t) to get
D−(1−α)
0
Gλi,τ,m
α,α (t) = D−(1−α)
0
∞
j=0
m + j
j
λj
i
(t − (m + j)τ)(m+ j)α+α−1
((m + j)α + α)
h(t − (m + j)τ)
=
∞
j=0
m + j
j
λj
i
D−(1−α)
(m+ j)τ
(t − (m + j)τ)(m+ j)α+α−1
((m + j)α + α)
h(t − (m + j)τ)
=
∞
j=0
m + j
j
λj
i
(t − (m + j)τ)(m+ j)α
((m + j)α + 1)
h(t − (m + j)τ) = Gλi,τ,m
α,1
(t)
(for the fractional power rule see [5]).
Furthermore, we employ the condition φ(t) ∈ L1([−τ, 0]) and Lemma 2 to estimate the influence of the initial function as
0
−τ
Gλi,τ,m
α,α (t − τ − u)φ(u)du ≤ sup
t−τ<u<t
|Gλi,τ,m
α,α (u)|
0
−τ
|φ(u)|du ≤
⎧
⎪⎨
⎪⎩
K1t−α−1
, λi ∈ Sα,τ ,
K2tmα+α−1
, λi = 0,
K3tm
, λi ∈ ∂Sα,τ \ {0},
K4tm
exp{|s1|(t − mτ)}, λi /∈ clSα,τ
for large t, where K1, K2, K3, K4 are suitable positive real constants. Note that if algebraic and geometric multiplicities of λi are
equal, then m = 0. Consequently, applying Lemma 2 to G
λi,τ,m
α,1
(t) we get the assertion of Theorem 2.
We point out that the proof of Theorem 2 actually enables us to specify the asymptotic behaviour of solutions y(t) of (2.1) also
in the unstable case. We can summarise it into the following
Theorem 5. Let A ∈ Rd×d, 0 < α < 1 and τ > 0. Let λi be all distinct eigenvalues of A (i = 1, . . . , n) and let the condition of
Theorem 2 (ii) be not satisfied (i.e. the zero solution of (2.1) is not stable). Then solutions y(t) of (2.1) admit three types of
asymptotics:
J. ˇCermák et al. / Commun Nonlinear Sci Numer Simulat 31 (2016) 108–123 119
(i) Let λ1 = 0 be the zero eigenvalue of A with algebraic multiplicity greater than geometric one and let p1 be the maximal size of
Jordan blocks corresponding to λ1. Furthermore, let λi ∈ Sα,τ for all i = 2, . . . , n. Then
y(t) ∼ t(p1−1)α as t → ∞ for any solution y (t) of (2.1).
(ii) Let λi (i = 1, . . . , ≤ n) be nonzero eigenvalues of A lying on ∂Sα,τ with algebraic multiplicity greater than geometric one and
let pi be the maximal size of Jordan blocks corresponding to λi (i = 1, . . . , ). Furthermore, let λi ∈ Sα,τ for all i = + 1, . . . , n
provided < n and p = max (p1, . . . , p ). Then
y(t) ∼sup tp−1
as t → ∞ for any solution y (t) of (2.1).
(iii) Let λi (i = 1, . . . , ≤ n) be eigenvalues of A located outside cl(Sα,τ ) and let s1 be a zero of (5.2) such that (s1) = max ( (s) :
s is a zero of (5.2) for some λ = λi, i = 1, . . . , ). Furthermore, let λj, j ∈ L ⊂ {1, . . . , } be eigenvalues of A such that (5.2) with
λ = λj has the zero s1 and let p be the maximal size of Jordan blocks corresponding to λj, j ∈ L. Then
y(t) ∼sup tp−1
exp{ (s1)t} as t → ∞ for any solution y (t) of (2.1).
7. Some consequences
In this section, we consider the system (2.1) in the planar form, i.e. when d = 2 and reformulate the stability conditions of
Theorem 2 quite explicitly in terms of trace of A ( tr A) and determinant of A (|A|). If d = 2, then the characteristic equation of A
becomes
λ2
− (tr A)λ + |A| = 0.
In such a case, an explicit reformulation of conditions of Theorem 2 is connected with tedious, but essentially straightforward
calculations. On this account, we omit these technical procedures and present only the relevant conclusion on the asymptotic
stability property of the zero solution.
Corollary 1. Let A ∈ R2×2, 0 < α < 1 and τ > 0. The zero solution of (2.1) is asymptotically stable if and only if
0 < τ2α|A| < (π − απ/2)2α
and
−
τ
π − απ/2
α
|A| −
π − απ/2
τ
α
< tr A < 2|A|1/2
cos τ|A|1/(2α) +
απ
2
.
Remark 6. The previous conditions enable various interpretations with respect to parameters tr A, |A|, α and τ. In particular, for
a changing τ (and with the remaining parameters being fixed), we get the following consequence:
If tr A ≤ −2|A|1/2, then the zero solution of the planar system (2.1) is asymptotically stable if and only if τ < τ∗ where
τ∗
= (π − απ/2)
−tr A − (tr A)2
− 4|A|
1/2
2|A|
1/α
.
Similarly, if tr A ≥ −2|A|1/2, then the zero solution of the planar system (2.1) is asymptotically stable if and only if τ < τ∗∗ where
τ∗∗
= arccos
tr A
2|A|1/2
− απ/2 |A|1/(2α).
From this viewpoint, the values τ∗ and τ∗∗ represent stability switches for the planar system (2.1), i.e. the critical values of a
delay τ when (2.1) loses its asymptotic stability property (for similar results on related integro-differential systems we refer to
[18]). Notice that the largest value of a delay τ, when asymptotic stability of the zero solution of (2.1) turns into instability, can
be achieved for tr A = −2|A|1/2. In this case, both the values τ∗ and τ∗∗ become
τ∗∗∗
=
π − απ/2
|A|1/(2α)
.
In other words, if tr A = −2|A|1/2, then the zero solution of the planar system (2.1) is asymptotically stable if and only if τ < τ∗∗∗.
For a better clarity, the conditions of Corollary 1 are depicted in Fig. 5.
Example 1. As an illustration of Corollary 1, we consider Duffing’s model of an unforced nonlinear oscillator
y1(t) = y2(t),
y2(t) = y1(t) − (y1(t))3
+ δy2(t)
(7.1)
depending on the damping parameter δ (we admit that δ is arbitrary real number). Note that involvement of other standard
parameters to this model has no contribution to our discussion and therefore we omit them.
Local stability analysis of (7.1) usually originates from the linearisation method. Application of this method to the nonlinear
system (7.1) along its equilibrium points (0, 0) and ( ± 1, 0) and analysis of the corresponding Jacobi matrices imply that the
120 J. ˇCermák et al. / Commun Nonlinear Sci Numer Simulat 31 (2016) 108–123
Fig. 5. The relationship between tr A and delay τ (with the fixed parameters α = 0.6 and |A| = 0.5).
Fig. 6. The norm of the solution y(t) of (7.3) for the values α = 0.25, τ = 0.35, δ = 0.01 and the initial function φ(t) ≡ (1, 1)T
on [−0.35, 0].
equilibrium point (0, 0) is not stable, while the equilibrium points ( ± 1, 0) are asymptotically stable for δ < 0, stable (but not
asymptotically) for δ = 0 and unstable for δ > 0.
If we replace the conventional derivatives in (7.1) by the fractional ones and the current time t on the right-hand side of (7.1)
by the delayed time t − τ, then we get the fractional delay Duffing’s model
Dα
0 y1(t) = y2(t − τ),
Dα
0 y2(t) = y1(t − τ) − (y1(t − τ))3
+ δy2(t − τ)
where 0 < α < 1 and τ > 0. This system has the same equilibrium points (0, 0) and ( ± 1, 0). The corresponding linearised models
along these points are
Dα
0 y1(t) = y2(t − τ),
Dα
0 y2(t) = y1(t − τ) + δy2(t − τ)
(7.2)
J. ˇCermák et al. / Commun Nonlinear Sci Numer Simulat 31 (2016) 108–123 121
Fig. 7. The norm of the solution y(t) of (7.3) for the values α = 0.5, τ = 0.35, δ = 0.01 and the initial function φ(t) ≡ (1, 1)T
on [−0.35, 0].
Fig. 8. The norm of the solution y(t) of (7.3) for the values α = 0.25, τ = 0.2, δ = 0.01 and the initial function φ(t) ≡ (1, 1)T
on [−0.2, 0].
and
Dα
0 y1(t) = y2(t − τ),
Dα
0 y2(t) = −2y1(t − τ) + δy2(t − τ),
(7.3)
respectively (for a linearisation theorem for fractional dynamical systems we refer to [19] and for its application to a two-term
fractional differential equation see also [20]). Applying Corollary 1 to (7.2), we get instability of the equilibrium point (0, 0)
due to |A| = −1 (hence, the involvement of fractional derivative order α and delay τ has no influence on its stability behaviour).
Regarding (7.3), the situation is more interesting. In this case, |A| = 2, tr A = δ and thus it is easy to rewrite the stability conditions
of Corollary 1. In particular, we can observe the following influence of parameters α and τ on the stability interval for the damping
parameter δ.
122 J. ˇCermák et al. / Commun Nonlinear Sci Numer Simulat 31 (2016) 108–123
If τ = 0, then the zero solution of (7.3) is asymptotically stable if and only if δ < 81/2cos (απ/2). If τ > 0, then the stability
interval for damping parameter δ becomes more restrictive, namely
−2
τ
π − απ/2
α
−
π − απ/2
τ
α
< δ < 81/2
cos τ21/(2α) + απ/2 (7.4)
where 0 < τ < (π − απ/2)/21/(2α) (see also Fig. 5 with respect to tr A = δ). In particular, if τ < (π/2 − απ/2)/21/(2α), then
(contrary to the classical model) the zero solution of (7.3) is asymptotically stable for δ = 0 as well as for appropriate positive
values of δ. The critical value of the stability switch τ∗∗∗ now equals (π − απ/2)/21/(2α).
To support these theoretical conclusions by a numerical experiment, we depict a norm of the vector solution y(t) of (7.3)
for three particular choices of entry parameters α, τ and δ. First let α = 0.25, τ = 0.35 and δ = 0.01. We can easily check that
this triplet does not satisfy the stability condition (7.4). If the value α is increased to α = 0.5 and the remaining parameters τ, δ
are unchanged, then (7.4) is already satisfied (it might be interesting to note that further increase of the value α leads again to
the loss of stability). Similarly, if the value τ is decreased to τ = 0.2 (with the original values α = 0.25 and δ = 0.01), then the
stability condition (7.4) is also met. The whole situation is depicted in Figs. 6–8 illustrating previous theoretical results. Figs. 7
and 8, corresponding to the asymptotically stable case, contain also asymptotic algebraic upper bounds (the dotted curves) for
the solutions y(t) of (7.3) following from Theorem 2 and Lemma 2.
8. Concluding remarks
In this paper, we have fully described the stability region for the fractional delay system (2.1), including the description of
asymptotics of its solutions. The utilised proof method was based on the Laplace transform method and analysis of zeros of
the corresponding characteristic equation. This approach has a wider usage and can be applied also in the study of some other
qualitative properties of FDDEs, e.g. their oscillatory behaviour (for some preliminary results in this direction we refer to [21]
whose results seem to be extendable just by our approach).
The assertion of Theorem 2 might be the starting point for stability analysis of other FDDEs. In the linear case, a natural
extension consists in the involvement of a non-delayed term on the right-hand side of (2.1), i.e. we can study the fractional delay
system
Dα
0 y(t) = Ay(t − τ) + By(t), t ∈ (0, ∞)
under various assumptions on matrices A, B (we emphasise that the problem of necessary and sufficient stability conditions for
this system with general d × d matrices A, B seems to be extremely complicated and it is still open even in the integer-order case
α = 1). In the nonlinear case, we have already outlined applicability of Theorem 2 and Corollary 1 in local stability investigations
of Duffing’s model of a nonlinear oscillator where we have described the influence of the fractional order α and the time delay
τ on its stability behaviour. These observations can be extended to local stability investigations of other nonlinear models which
enables to describe their dynamics in the neighbourhood of equilibrium points. Another potential applicability of our results
consists in the synchronisation process between a drive system (described via a nonlinear FDDE) and the corresponding response
system involving some control parameters. Then the synchronisation between both the systems is equivalent to the asymptotic
stability of the zero solution of the corresponding error system consisting of linear FDDEs (see, e.g. [9]). Finally, (2.1) may serve
as the test system either for numerical analysis of FDDEs (for a similar situation in the fractional non-delayed case we refer to
[22,23]), or for stability investigations of FDDEs via other methods (for basic principles of Lyapunov methods for FDDEs see, e.g.
[24]).
Acknowledgements
The authors are grateful to referees for reading this paper, suggestions and recommendations which considerably helped to
improve its content. The research was supported by the grant P201/11/0768 of the Czech Science Foundation and by the projects
CZ.1.07/2.3.00/30.0039 and FSI-S-14-2290 of Brno University of Technology.
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Appendix A. Lower-order one-term FDDS (CNSNS, 2016) 64
Appendix B
Paper on higher-order one-term
FDDS [10] (EJDE, 2019)
We do not find direct technical generalizations of previous papers particularly interesting,
which likely led us to postpone the work on higher-order one-term FDDS as
it seemed like a simple follow-up on [6]. However, our hesitation proved unnecessary.
While the stability results were indeed expectedly straightforward generalizations of
our previous findings, [10] (co-author: J. Čermák; my author’s share 50 %) shifted
our focus towards the oscillatory properties - a challenge introduced by higher-order
systems.
In this paper, we conducted a deeper analysis of the locations of characteristic
roots depending on location of eigenvalues. Unlike common practice, we were not
only interested in the case of negative real parts. We detailed the occurrence and
conditions of roots with positive real parts, including their number. That started
our interest in the properties of unbounded solutions, which we revisited also in
subsequent papers.
Ultimately, [10] addresses FDDS of all positive non-integer orders, revealing predominantly
non-oscillatory behaviour. To better discuss the mechanism by which
initial conditions influence the oscillatory and stability properties of the given solution,
we introduced the terms major and n-minor solutions. That allowed us to
explore the effects of initial conditions more comprehensively.
– 65 –
Electronic Journal of Differential Equations, Vol. 2019 (2019), No. 33, pp. 1–15.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
OSCILLATORY AND ASYMPTOTIC PROPERTIES OF
FRACTIONAL DELAY DIFFERENTIAL EQUATIONS
JAN ˇCERM´AK, TOM´AˇS KISELA
Abstract. This article discusses the oscillatory and asymptotic properties
of a test delay differential system involving a non-integer derivative order.
We formulate corresponding criteria via explicit necessary and sufficient conditions
that enable direct comparisons with the results known for classical
integer-order delay differential equations. In particular, we shall observe that
oscillatory behaviour of solutions of delay system with non-integer derivatives
embodies quite different features compared to the classical results known from
the integer-order case.
1. Introduction and preliminaries
Basic qualitative properties of the delay differential equation
y (t) = Ay(t − τ), t ∈ (0, ∞), (1.1)
where A is a constant real d × d matrix and τ > 0 is a constant real lag, are
well described in previous numerous investigations. While stability and asymptotic
properties of (1.1) were reported in [8], answers to various oscillation problems
regarding (1.1) were surveyed in [7].
A crucial role in these investigations was played by the associated characteristic
equation
det(sI − A exp{−sτ}) = 0, (1.2)
where I is the identity matrix. More precisely, appropriate properties of (1.1) were
first described via location of all roots of (1.2) in a specific area of the complex
plane. Then, efficient criteria guaranteeing such root locations were formulated in
terms of conditions imposed directly on the eigenvalues of A.
We recall some of relevant statements (reformulated in the above mentioned
sense) along with their consequences to the scalar case when (1.1) becomes
y (t) = ay(t − τ), t ∈ (0, ∞) (1.3)
where a is a real number. Since we are primarily interested in discussions of oscillatory
properties of appropriate fractional extensions of (1.1), we first state (see [7])
oscillation conditions for (1.1) (as it is customary, we say that a solution of (1.1) is
oscillatory if every its component has arbitrarily large zeros; otherwise the solution
is called non-oscillatory).
2010 Mathematics Subject Classification. 34K37, 34A08, 34K11, 34K20.
Key words and phrases. Fractional delay differential equation; oscillation;
asymptotic behaviour.
c 2019 Texas State University.
Submitted March 20, 2018. Published February 22, 2019.
1
2 J. ˇCERM´AK, T. KISELA EJDE-2019/33
Theorem 1.1. Let A ∈ Rd×d
and τ ∈ R+
. Then the following statements are
equivalent:
(a) All solutions of (1.1) oscillate;
(b) The characteristic equation (1.2) has no real roots;
(c) A has no real eigenvalues in [−1/(τe), ∞).
Corollary 1.2. Let a ∈ R and τ ∈ R+
. All solutions of (1.3) oscillate if and only
if
a < −
1
τe
.
As we shall see later, oscillatory properties of the corresponding fractional delay
system are closely related to convergence of all its solutions to the zero solution. In
the first-order case (1.1), this property was characterized in [8] via
Theorem 1.3. Let A ∈ Rd×d
and τ ∈ R+
. Then the following statements are
equivalent:
(a) Any solution y of (1.1) tends to zero as t → ∞;
(b) The characteristic equation (1.2) has all roots with negative real parts;
(c) All eigenvalues λi (i = 1, . . . , d) of A satisfy
τ|λi| < | arg(λi)| − π/2 .
Moreover, the convergence of y to zero is of exponential type.
Remark 1.4. The condition (c) can be equivalently expressed via the requirement
that all eigenvalues λi (i = 1, . . . , d) of A have to be located inside the region
bounded by the curve
(λ) = ω cos(ωτ), (λ) = −ω sin(ωτ), −
π
2τ
≤ ω ≤
π
2τ
in the complex plane.
Corollary 1.5. Let a ∈ R and τ ∈ R+
. Any solution y of (1.3) tends to zero as
t → ∞ if and only if
−
π
2τ
< a < 0 .
Extensions of previous results to the n-th order equation (n is a positive integer)
y(n)
(t) = Ay(t − τ), t ∈ (0, ∞) (1.4)
yield different conclusions. In this case, the characteristic equation becomes
det(sn
I − A exp{−sτ}) = 0 . (1.5)
If n ≥ 2, then there is no analogue to Theorem 1.3. More precisely, the convergence
of all solutions of (1.4) to zero is not possible whenever n ≥ 2 (see, e.g. [6]).
Regarding oscillatory properties of (1.4), equivalency of conditions (a) and (b) (with
(1.2) replaced by (1.5)) of Theorem 1.1 remains preserved, but their conversion into
an explicit form depends on parity of n (see [7]).
The main goal of this article is to discuss these oscillatory and related asymptotic
properties of (1.1) with respect to their possible extension to the fractional delay
differential equation
Dα
0 y(t) = Ay(t − τ), t ∈ (0, ∞) (1.6)
EJDE-2019/33 OSCILLATIONS FOR FRACTIONAL DELAY EQUATIONS 3
where α > 0 is a real scalar and the symbol Dα
0 is the Caputo derivative of order
α introduced in the following way: First let y be a real scalar function defined on
(0, ∞). For a positive real γ, the fractional integral of y is given by
D−γ
0 y(t) =
t
0
(t − ξ)γ−1
Γ(γ)
y(ξ)dξ, t ∈ (0, ∞)
and, for a positive real α, the Caputo fractional derivative of y is given by
Dα
0 y(t) = D
−( α −α)
0
d α
dt α
y(t) , t ∈ (0, ∞)
where · means the upper integer part. As it is customary, we put D0
0y(t) = y(t)
(for more on fractional calculus, see, e.g. [10, 15]). If y is a real vector function, the
corresponding fractional operators are considered component-wise (similarly, if y
is a complex-valued function, then these fractional operators are introduced for its
real and imaginary part separately). We add that the initial conditions associated
to (1.6) are
y(t) = φ(t), t ∈ [−τ, 0] , (1.7)
lim
t→0+
y(j)
(t) = φj, j = 0, . . . , α − 1 (1.8)
where all components of φ are absolutely Riemann integrable on [−τ, 0] and φj are
real scalars. In the frame of our oscillatory and asymptotic discussions on (1.6), we
are going not only to extend previous results to (1.6) but also discuss a dependence
of relevant conditions on changing derivative order α (with a special attention to
the case when α is crossing integer values).
The structure of this paper is following: Section 2 recalls some related special
functions as well as the characteristic equation associated with (1.6). Some asymptotic
expansions of the studied special functions are described as well. In Section
3, we discuss in detail distribution of roots of the characteristic equation in specific
areas of the complex plane. Using these auxiliary statements, Sections 4 and 5 formulate
a series of results describing oscillation and asymptotic properties of (1.6)
in the vector and scalar case. More precisely, Section 4 presents analogues of Theorems
1.1 and 1.3, and Section 5 contains some additional oscillation results in the
scalar case. Discussions on non-consistency of the obtained results with the above
recalled classical properties of (1.1) and (1.3) are subject of Section 6 concluding
the paper.
2. Special functions and their properties
In this section, we recall and extend some notions and formulae introduced in
[3] in the frame of stability analysis of (1.6) with 0 < α < 1. As we shall see
later, these tools turn out to be very useful also in oscillatory investigations of (1.6)
with arbitrary real α > 0. Since the proofs of auxiliary statements stated below
are (essentially) analogous to the proofs of appropriate assertions from [3], we omit
them.
In the sequel, the symbols L and L−1
denote the Laplace transform and inverse
Laplace transform of appropriate functions, respectively.
Definition 2.1. Let A ∈ Rd×d
, let I be the identity d×d matrix and let α, τ ∈ R+
.
The matrix function R : R → Cd×d
given by
R(t) = L−1
(sα
I − A exp{−sτ})−1
(t) (2.1)
4 J. ˇCERM´AK, T. KISELA EJDE-2019/33
is called the fundamental matrix solution of (1.6).
Theorem 2.2. Let A ∈ Rd×d
, α, τ ∈ R+
and let R be the fundamental matrix
solution of (1.6). Then the solution y of (1.6)–(1.8) is given by
y(t) =
α −1
j=0
Dα−j−1
0 R(t)φj +
0
−τ
R(t − τ − u)Aφ(u)du.
Remark 2.3. Theorem 2.2 along with Definition 2.1 imply that the poles of the
Laplace image of solution of (1.6) coincide with roots of
det(sα
I − A exp{−sτ}) = 0, equivalently
n
i=1
(sα
− λi exp{−sτ})ni
= 0 , (2.2)
where λi (i = 1, . . . , n) are distinct eigenvalues of A and ni are their algebraic
multiplicities. This confirms the well-known fact that (2.2) is the characteristic
equation associated to (1.6) (see, e.g. [5, 9, 11]).
The following notion of a generalized delay exponential function plays an important
role in description of asymptotic expansions of the fundamental matrix
solution of (1.6).
Definition 2.4. Let λ ∈ C, η, β, τ ∈ R+
and m ∈ Z+
∪ {0}. The generalized delay
exponential function (of Mittag-Leffler type) is introduced via
Gλ,τ,m
η,β (t) =
∞
j=0
m + j
j
λj
(t − (m + j)τ)η(m+j)+β−1
Γ(η(m + j) + β)
h(t − (m + j)τ)
where h is the Heaviside step function.
The relationship between the fundamental matrix solution R and the generalized
delay exponential functions Gλ,τ,m
η,β can be specified via the following lemma.
Lemma 2.5. The fundamental matrix solution (2.1) can be expressed as R(t) =
T−1
G(t)T, where T is a regular matrix and G is a block diagonal matrix with uppertriangular
blocks Bj given by
Bj(t) =








Gλi,τ,0
α,α (t) Gλi,τ,1
α,α (t) Gλi,τ,2
α,α (t) · · · G
λi,τ,rj −1
α,α (t)
0 Gλi,τ,0
α,α (t) Gλi,τ,1
α,α (t) · · · G
λi,τ,rj −2
α,α (t)
0 0 Gλi,τ,0
α,α (t) · · · G
λi,τ,rj −3
α,α (t)
...
...
...
...
...
0 0 0 · · · Gλi,τ,0
α,α (t)








,
where j = 1, . . . , J (J ∈ Z+
) and rj is the size of the corresponding Jordan block of
A.
As a next key auxiliary result, we describe asymptotic behaviour of Gλ,τ,m
η,β func-
tions.
Lemma 2.6. Let λ ∈ C, α ∈ R+
\ Z+
, β, τ ∈ R+
and m ∈ Z+
∪ {0}. Further, let
si (i = 1, 2, . . . ) be the roots of
sα
− λ exp{−sτ} = 0 (2.3)
with ordering (si) ≥ (si+1) (i = 1, 2, . . . ; in particular, s1 is the rightmost root).
EJDE-2019/33 OSCILLATIONS FOR FRACTIONAL DELAY EQUATIONS 5
(i) If λ = 0, then
G0,τ,m
α,β (t) =
(t − mτ)mα+β−1
Γ(mα + β)
h(t − mτ).
(ii) If λ = 0, then
Gλ,τ,m
α,β (t) =
∞
i=1
m·ki
j=0
aij(t − mτ)j
exp{si(t − mτ)} + Sλ,τ,m
α,β (t) ,
where ki is a multiplicity of si, aij are suitable nonzero complex constants
(j = 0, . . . , mki, i = 1, 2, . . . ) and the term Sλ,τ,m
α,β has the algebraic asymptotic
behaviour expressed via
Sλ,τ,m
α,β (t) =
(−1)m+1
λm+1Γ(β − α)
(t + τ)β−α−1
+
(−1)m+1
(m + 1)
λm+2Γ(β − 2α)
(t + 2τ)β−2α−1
+ O(tβ−3α−1
) as t → ∞.
3. Distribution of characteristic roots
The aim of this section is to analyse (2.2) with respect to existence of its real
roots as well as number of its roots with positive real parts. Doing this, it is enough
to consider its partial form (2.3).
First, we characterize the set of all roots of (2.3) in terms of their magnitudes
and arguments (we assume here λ = 0, i.e. s = 0). Using the goniometric forms of
s and λ we obtain that (2.3) is equivalent to
|s|α
cos[α arg(s)] − |λ| exp{−|s|τ cos[arg(s)]} cos[arg(λ) − |s|τ sin(arg(s))]
= 0,
(3.1)
|s|α
sin[α arg(s)] − |λ| exp{−|s|τ cos[arg(s)]} sin[arg(λ) − |s|τ sin(arg(s))]
= 0.
(3.2)
To solve (2.3), we consider (3.1)–(3.2) as a system with unknowns |s| and arg(s).
If α arg(s) = 1π for some integer 1, then arg(λ) − |s|τ sin[arg(s)] = 2π for some
integer 2 and (3.1) yields
|s|α
(−1) 1
− |λ| exp{−|s|τ cos[arg(s)]}(−1) 2
= 0,
i.e.
|s|α
= (−1) |λ| exp{−|s|τ cos[arg(s)]} = 0 for some integer . (3.3)
Thus (3.1)–(3.2) can be reduced to
α arg(s) − arg(λ) − |s|τ sin[arg(s)] = 2kπ for some integer k, (3.4)
|s|α
= |λ| exp{−|s|τ cos[arg(s)]}. (3.5)
If α arg(s) = 1π for any integer 1, then arg(λ) − |s|τ sin[arg(s)] = 2π for any
integer 2 and division (3.1) over (3.2) yields
α arg(s) = |λ| exp{−|s|τ cos[arg(s)]} + π for some integer .
This, after substitution into (3.1), yields (3.3). Now, the same argumentation as
above shows equivalency of (2.3) and (3.4)–(3.5).
Using the previous process, we can derive the following characterization of possible
real roots of (2.3).
6 J. ˇCERM´AK, T. KISELA EJDE-2019/33
Proposition 3.1. Let λ ∈ C and α, τ ∈ R+
.
(i) The characteristic equation (2.3) has a positive real root if and only if λ is
a positive real. This root is simple, unique and it is the rightmost root of
(2.3).
(ii) The characteristic equation (2.3) has a negative real root if and only if
0 < |λ| ≤ (
α
τe
)α
and arg(λ) = (α − 2k)π for some k ∈ Z .
More precisely, if
0 < |λ| = (
α
τe
)α
and arg(λ) = (α − 2k)π for some k ∈ Z ,
then s1,2 = −α/τ is double and the rightmost root of (2.3) (remaining roots
of (2.3) are not real). If
0 < |λ| <
α
τe
α
and arg(λ) = (α − 2k)π for some k ∈ Z ,
then (2.3) has a couple of simple real negative roots, the greater of them
being rightmost (remaining roots of (2.3) are not real).
(iii) The characteristic equation (2.3) has the zero root if and only if λ = 0.
Furthermore, using (3.4)–(3.5) we can specify the distribution of characteristic
roots of (2.3) with respect to the imaginary axis. Before doing this, we introduce
the following areas in the complex plane.
For real parameters 0 < α < 2 and τ > 0, we define the set Q0(α, τ) of all
complex λ such that
| arg(λ)| >
απ
2
and |λ| <
| arg(λ)| − απ
2
τ
α
.
Further, for any positive integer m and real parameters 0 < α < 4m + 2 and τ > 0,
we define the sets Qm(α, τ) of all complex λ such that either
απ
2
− 2mπ < | arg(λ)| ≤
απ
2
− (2m − 2)π and |λ| <
| arg(λ)| − απ
2 + 2mπ
τ
α
,
or | arg(λ)| > απ
2 − 2mπ and
| arg(λ)| − απ
2 + (2m − 2)π
τ
α
< |λ| <
| arg(λ)| − απ
2 + 2mπ
τ
α
.
We add that the sets Qm(α, τ) (m = 0, 1, . . . ) are defined to be empty whenever
α ≥ 4m + 2.
Now, we can describe the location of the roots of (2.3) with respect to the
imaginary axis in terms of the sets Qm(α, τ) (we utilize here the standard notation
∂[Qm(α, τ)] for their boundaries).
Proposition 3.2. Let λ ∈ C and α, τ ∈ R+
. Then there exist just m (m = 0, 1, . . . )
characteristic roots of (2.3) with a positive real part (while remaining roots have
negative real parts) if and only if λ ∈ Qm(α, τ). Moreover, (2.3) has a root with the
zero real part if λ ∈ ∂[Qm(α, τ)] for some m = 0, 1, . . . .
The appropriate regions Qm(α, τ) are depicted in Figures 1 and 2.
EJDE-2019/33 OSCILLATIONS FOR FRACTIONAL DELAY EQUATIONS 7
Im(λ)
Re(λ)
Q (α,τ)0
Q (α,τ)1
Q (α,τ)2
Q (α,τ)3
Im(λ)
Re(λ)
Q (α,τ)2
Q (α,τ)3
Q (α,τ)0
Q (α,τ)1
Q (α,τ)4
Figure 1. α = 0.4 and τ = 1 (left). α = 1.1 and τ = 1 (right)
Im(λ)
Re(λ)
Q (α,τ)3
Q (α,τ)2
Q (α,τ)1
Q (α,τ)4
Im(λ)
Im(λ)
Re(λ)
Re(λ)
Q (α,τ)1
Q (α,τ)3
Q (α,τ)2
Q (α,τ)4
Figure 2. α = 2.1 and τ = 1 (left). α = 3.1 and τ = 1 (right)
Proof. We start with the proof of Proposition 3.1 and consider the characterization
of roots s of (2.3) via (3.4)–(3.5). Obviously, (2.3) has a positive real root if
arg(λ) = 0 (i.e. λ is a positive real). In this case, the characteristic function
F(s) = sα
− λ exp{−sτ}
is strictly increasing for all s ≥ 0 with F(0) = −λ < 0 and F(∞) = ∞, hence
there is a unique positive real root s1 of (2.3). To show its dominance, we consider
remaining roots si of (2.3) with a positive real parameter λ. Then (3.5) yields
(s1)α
= λ exp{−s1τ}, |si|α
= λ exp{−|si|τ cos[arg(si)]}.
From here, we obtain
s1
|si|
α
= exp{(−s1 + |si| cos[arg(si)])τ} . (3.6)
8 J. ˇCERM´AK, T. KISELA EJDE-2019/33
Assume that s1 is not the rightmost root of (2.3), i.e. |si| cos[arg(si)] ≥ s1 for some
root si of (2.3). Then
s1
|si|
< 1 and exp{(−s1 + |si| cos[arg(si)])τ} ≥ 1
which contradicts (3.6). This proves Proposition 3.1(i).
Similarly, (3.4)–(3.5) imply that (2.3) has a negative real root s if and only if
arg(λ) = (α − 2k)π for some k ∈ Z
and
|s|α
= |λ| exp{|s|τ}.
Put r = |s| and G(r) = rα
− |λ| exp{rτ}, r ≥ 0. Then G(0) = −|λ| < 0, G(∞) =
−∞ and G is increasing in (0, r∗
) and decreasing in (r∗
, ∞) for a suitable r∗
> 0.
Thus G has (one or two) positive roots if and only if G(r∗
) ≥ 0. In particular, G
has a unique positive root r∗
if and only if G(r∗
) = G (r∗
) = 0, i.e.
(r∗
)α
− |λ| exp{r∗
τ} = α(r∗
)α−1
− |λ|τ exp{r∗
τ} = 0 .
From here, we obtain
r∗
=
α
τ
and |λ| =
α
τe
α
.
Obviously, if
|λ| <
α
τe
α
,
then G has two real positive roots r1 < r2. We show that s1 = −r1 is the rightmost
root of (2.3), i.e s1 > |si| cos[arg(si)] for all remaining roots si (i = 2, 3, . . . ) of
(2.3). Indeed, by (3.5),
|s1|α
= |λ| exp{|s1|τ} and |si|α
= |λ| exp{−|si|τ cos[arg(si)]} .
Then |s1| < |si|, i.e. |s1| + |si| cos[arg(si)] < 0. Analogously we can show the
dominance of a double real root s1,2 (if exists). This proves Proposition 3.1 (ii).
The assertion of Proposition 3.1(iii) is trivial.
Now, we show the validity of Proposition 3.2. Since the case of real characteristic
roots of (2.3) has been discussed previously, we first search the roots s with 0 <
arg(s) ≤ π/2. Then (3.4)–(3.5) can be reduced to
|s| =
arg(λ) − α arg(s) + 2kπ
τ sin[arg(s)]
, (3.7)
arg(λ) − α arg(s) + 2kπ
τ sin[arg(s)]
α
− |λ| exp (− arg(λ)
+ α arg(s) − 2kπ) cotan[arg(s)] = 0 .
(3.8)
We denote the left-hand side of (3.8) by Hk = Hk(arg(s)). Then
Hk(0+
) = ∞, Hk(π/2) =
arg(λ) − απ/2 + 2kπ
τ
α
− |λ|
and Hk(arg(s)) decreases as arg(s) increases from 0 to π/2. This implies that (3.7)–
(3.8) has just m couples of solutions with |s| > 0 and 0 < arg(s) ≤ π/2 if and only
if either
απ
2
− 2mπ < arg(λ) ≤
απ
2
− (2m − 2)π and Hm(π/2) > 0 ,
or
arg(λ) >
απ
2
− 2mπ and Hm(π/2) > 0 > Hm−1(π/2) .
EJDE-2019/33 OSCILLATIONS FOR FRACTIONAL DELAY EQUATIONS 9
If −π/2 ≤ arg(s) < 0, then we obtain the same conclusion with arg(λ) replaced by
− arg(λ). This implies the main part of the assertion. The supplement on existence
of purely imaginary roots of (2.3) follows from continuous dependence of roots s on
parameter λ. Alternatively, it can be obtained via the standard D-decomposition
method.
4. Main results
In this section, we derive and formulate fractional-order analogues to Theorems
1.1 and 1.3.
Theorem 4.1. Let A ∈ Rd×d
, α ∈ R+
\ Z+
and τ ∈ R+
. Then the following
statements are equivalent:
(a) All non-trivial solutions of (1.6) are non-oscillatory;
(b) The characteristic equation (2.2) admits only real roots or roots with a
negative real part;
(c) A has all eigenvalues lying in Q0(α, τ) ∪ (Q1(α, τ) ∩ R) ∪ {0}.
Proof. Theorem 2.2 and Lemma 2.5 imply that every solution of (1.6)–(1.8) can be
expressed as
y(t) = T−1
α −1
j=0
Dα−j−1
0 G(t)Tφj + T−1
0
−τ
G(t − τ − u)JTφ(u)du , (4.1)
where G is a matrix function introduced in Lemma 2.5, J is a Jordan form of
the system matrix A and T is the corresponding regular projection matrix, i.e.
A = TJT−1
. Employing (4.1) and Lemma 2.5, we can see that every component
of y is a linear combination of terms derived from elements of G. We distinguish
two cases with respect to (non)zeroness of eigenvalues λi of A.
First, let λi = 0 for all i = 1, . . . , n (n being the number of distinct eigenvalues
of A). Then the elements of matrices Dα−j−1
0 G(t) (j = 0, . . . , α − 1) can be
asymptotically expanded via the relation
Dα−j−1
0 Gλi,τ,m
α,α (t) = Gλi,τ,m
α,j+1 (t)
=
N
w=1
mkw
=0
t exp{swt}bw, 1 −
mτ
t
exp{−swmτ}
+ tj−α (−1)m+1
(1 + τ/t)j−α
λm+1
i Γ(j − α + 1)
+ O(tj−2α
) as t → ∞ ,
(4.2)
where sw (w = 1, 2, . . . , N) are roots of (2.3) with the largest real parts ordered as
(sw) ≥ (sw+1), N is any positive integer satisfying (sN ) < 0, kw is multiplicity
of sw and bw, are suitable real constants (see ai,j in Lemma 2.6(ii)). Similarly, the
10 J. ˇCERM´AK, T. KISELA EJDE-2019/33
elements of the matrix
0
−τ
G(t − τ − u)JTφ(u)du have the expansions
0
−τ
Gλi,τ,m
α,α (t − τ − u)ˆφp
(u)du
=
N
w=1
mkw
=0
t exp{swt}cw, λi
0
−τ
1 −
(m + 1)τ
t
−
u
t
e−sw((m+1)τ+u) ˆφp
(u)du
+ t−α−1
0
−τ
(−1)m+1
(m + 1)(1 + τ/t − u/t)−α−1
λm+1
i Γ(−α)
ˆφp
(u)du + O(t−2α−1
)
(4.3)
as t → ∞, where ˆφp
(u) is pth row of the vector JTφ(u) and cw, are suitable real
constants (see ai,j in Lemma 2.6(ii)).
If λi = 0 for some i = 1, . . . , n, then the appropriate analogues of (4.2)–(4.3)
involve only algebraic terms (see Lemma 2.6(i)). Now, we can prove the presented
equivalencies:
(a)⇔(b): The property (a) holds if and only if, for any choice of φ, the dominating
terms involved in (4.2) and (4.3) are non-oscillatory. We can see that all the
algebraic terms from (4.2) and (4.3) are non-oscillatory and eventually dominating
with respect to all exponential terms with negative real parts of their arguments.
Contrary, an exponential term is eventually dominating provided its argument has
a non-negative real part. Clearly, if such a case does occur, the solution y of (1.6) is
non-oscillatory only if the imaginary parts of the corresponding arguments are zero.
By (4.2) and (4.3), the discussed arguments of the exponential terms are expressed
via roots of (2.2), which yields equivalency of (a) and (b).
(b)⇔(c): This equivalency follows immediately from Propositions 3.1 and 3.2.
In the scalar case, when (1.6) becomes
Dα
0 y(t) = ay(t − τ), t ∈ (0, ∞) , (4.4)
a being a real scalar, we obtain the following explicit characterization of nonexistence
of a non-trivial oscillatory solution.
Corollary 4.2. Let a ∈ R, α ∈ R+
\ Z+
and τ ∈ R+
. All non-trivial solutions y
of (4.4) are non-oscillatory if and only if
0 < α < 2 and −
(2 − α)π
2τ
α
< a <
(4 − α)π
2τ
α
,
or
2 < α < 4 and 0 < a <
(4 − α)π
2τ
α
.
Remark 4.3. In the first-order case, the value a = −1/(τe) is of a particular importance:
crossing this value, the (negative) real roots of the associated characteristic
equation disappear and all solutions of (1.3) become oscillatory for a < −1/(τe).
In the fractional-order case, the (negative) real roots disappear for a < −(α/(τe))α
.
However, such roots have no impact on oscillatory behaviour of the solutions of (4.4)
because the exponential terms with negative arguments involved in the formulae
(4.1)–(4.3) are eventually suppressed by algebraic terms.
By Theorem 4.1, if all roots of (2.2) have negative real parts, then all non-trivial
solutions of (1.6) are non-oscillatory. Therefore, we give an explicit characterization
of this assumption and thus provide a fractional-order analogue to Theorem 1.3.
EJDE-2019/33 OSCILLATIONS FOR FRACTIONAL DELAY EQUATIONS 11
Theorem 4.4. Let A ∈ Rd×d
and α, τ ∈ R+
. Then the following statements are
equivalent
(a) Any solution y of (1.6) tends to zero as t → ∞;
(b) The characteristic equation (2.2) has all roots with negative real parts;
(c) All eigenvalues λi (i = 1, . . . , d) of A are nonzero and satisfy
τ|λi|1/α
< |arg (λi)| − απ/2 .
Moreover, if α /∈ Z+
, then the convergence to zero is of algebraic type; more precisely,
for any solution y of (1.6) there exists a suitable integer j ∈ {0, . . . , α }
such that |y(t)| ∼ tj−α−1
as t → ∞ (the symbol ∼ stands for asymptotic equiva-
lency).
Proof. (a)⇔(b): If λi = 0 for some i = 1, . . . , d, then the appropriate analogues
of (4.2) and (4.3) yield that there is always a constant term involved in these
expansions (this constant is nonzero if φ0 is nonzero), hence the property (a) is
not true. Obviously, the property (b) cannot occur as well provided λi = 0 for
some i = 1, . . . , d. Thus, without loss of generality, we may assume λi = 0 for all
i = 1, . . . , d.
The statement (a) is valid if and only if (4.2) and (4.3) do not contain any terms
with a non-negative real part of the argument, which directly yields the equivalency
(see also [11]).
(b)⇔(c): It is a direct consequence of Proposition 3.2.
Consequently, since all the exponential terms in (4.2) and (4.3) have a negative
argument, they are suppressed by the algebraic terms. The presence of the term
behaving like tj−α−1
for j = 1, . . . , α as t → ∞ is determined by values φj−1.
If φj−1 = 0 for all j = 1, . . . , α , the integral term (4.3) becomes dominant. The
integrability of φ enables us to write
lim
t→∞
1
t−α−1
0
−τ
Gλi,τ,m
α,α (t − τ − u)ˆφp
(u)du
=
0
−τ
lim
t→∞
(−1)m+1
(m + 1)(1 + τ/t − u/t)−α−1
λm+1
i Γ(−α)
ˆφp
(u)du
= K
0
−τ
ˆφp
(u)du
for a suitable real K, therefore the integral term behaves like t−α−1
as t → ∞. This
completes the proof.
For the case of scalar equation (4.4), we obtain the following result.
Corollary 4.5. Let a ∈ R and α, τ ∈ R+
. All solutions y of (4.4) tend to zero if
and only if
α < 2 and −
(2 − α)π
2τ
α
< a < 0 .
In particular, an interesting link between Theorems 4.1 and 4.4 is provided by
the following assertion.
Corollary 4.6. Let A ∈ Rd×d
, α ∈ R+
\ Z+
and τ ∈ R+
. If (1.6) has a nontrivial
oscillatory solution, then it has also a solution which does not tend to zero
as t → ∞.
12 J. ˇCERM´AK, T. KISELA EJDE-2019/33
Remark 4.7. In fact, formulae (4.1)–(4.3) reveal that any non-trivial solution of
(1.6) tending to zero is non-oscillatory. Moreover, the solutions tending to zero
pose an algebraic decay (there is no solution with an exponential decay).
5. Other oscillatory properties of (4.4)
In the classical integer-order case, oscillation argumentation often uses the fact
that exp(swt) is a solution of (1.3) for any root sw of the corresponding characteristic
equation
s − a exp{−sτ} = 0 . (5.1)
In particular, if (5.1) admits a real root, then (1.3) has (via appropriate choice of φ)
a non-oscillatory solution. In the fractional-order case, no such a direct connection
for the influence study of characteristic roots of
sα
− a exp{−sτ} = 0 (5.2)
on the oscillatory behaviour of (4.4) is available. Nevertheless, as we can see from
(4.1)–(4.3), the exponential functions generated by characteristic roots of (5.2)
again play an important role in qualitative analysis of solutions of (4.4). Using
this fact, we are able to describe some oscillatory properties of (4.4) with respect
to asymptotic relationship between the studied solutions and the corresponding
exponential functions. To specify this relationship, we introduce the following asymptotic
classifications of solutions of (4.4).
Definition 5.1. Let a ∈ R and α, τ ∈ R+
. The solution y of (4.4) is called major
solution, if it satisfies the asymptotic relationship
lim sup
t→∞
y(t)
tk1 exp{s1t}
> 0 ,
where s1 is the rightmost root of (5.2) and k1 its algebraic multiplicity.
Definition 5.2. Let a ∈ R, α, τ ∈ R+
, sw (w = 1, 2, . . . ) be roots of (5.2) with
ordering (sw) ≥ (sw+1) and let kw (w = 1, 2, . . . ) be the corresponding algebraic
multiplicities. The solution y of (4.4) is called m-minor solution, if it satisfies the
asymptotic relationships
lim sup
t→∞
y(t)
tkm exp{smt}
= 0 and lim sup
t→∞
y(t)
tkm+1 exp{sm+1t}
> 0 .
Remark 5.3. The notions of the major and m-minor solutions are not just theoretical,
but such solutions can be constructively obtained via appropriate choice of
the initial function φ. For example, if s1 is simple with a non-negative real part,
then, by (4.1)–(4.3), the major solution occurs if φ meets the condition
α −1
j=0
φjb1,j + ac1,0
0
−τ
φ(u) exp{−s1(τ + u)}du = 0
where b1,j, c1,0 have the same meaning as in (4.2)–(4.3). Clearly, such a condition
is satisfied by infinitely many initial functions, e.g. by φ(u) = 1, φj = 0
(j = 1, . . . , α − 1) and φ0 = −ac1,0(1 − exp{−s1τ})/(b1,0s1). Similarly, m-minor
solution is characterized by the conditions
α −1
j=0
φjbw,j + acw,0
0
−τ
φ(u) exp{−sw(τ + u)}du = 0 for w = 1, . . . , m ,
EJDE-2019/33 OSCILLATIONS FOR FRACTIONAL DELAY EQUATIONS 13
α −1
j=0
φjbm+1,j + acm+1,0
0
−τ
φ(u) exp{−sm+1(τ + u)}du = 0
provided sw (w = 1, . . . , m+1) are simple roots and bw,j, cw,0, bm+1,j, cm+1,0 have
the same meaning as in (4.2)–(4.3).
Using the notions of major and m-minor solutions, we can formulate in a more
detail assertions revealing the relation between oscillatory properties of (4.4) and
location of roots of (5.2) in the complex plane.
Lemma 5.4. Let a ∈ R \ (Q0(α, τ) ∪ {0}), α ∈ R+
\ Z+
, τ ∈ R+
and let sw
(w = 1, 2, . . . ) be roots of (5.2) with ordering (sw) ≥ (sw+1). Then the major
solutions of (4.4) do not tend to zero and there exists M > 0 such that all m-minor
solutions of (4.4) are non-oscillatory and tend to zero as t → ∞ for all m ≥ M.
Furthermore, it holds:
(i) If a ≤ −((2 − α)π/(2π))α
for α < 2 or a < 0 for α > 2, then all major
solutions of (4.4) are oscillatory.
(ii) If α < 4 and 0 < a < ((4 − α)π/(2π))α
, then all non-trivial solutions of
(4.4) are non-oscillatory.
(iii) If α < 4 and a = ((4 − α)π/(2π))α
, then all major solutions of (4.4)
are non-oscillatory. Moreover, all 1-minor solutions are oscillatory and
bounded.
(iv) If a > ((4 − α)π/(2π))α
for α < 4 or a > 0 for α > 4, then all major
solutions of (4.4) are non-oscillatory. Moreover, all 1-minor solutions are
oscillatory and unbounded.
Proof. The first part of the assertion follows from the expansion of solution y of
(4.4) based on (4.2)–(4.3). By Proposition 3.2, the rightmost root s1 has a nonnegative
real part, therefore the major solutions involve, as a dominant term, an
exponential function which does not tend to zero. Using a technique similar to that
in Remark 5.3 we can always eliminate all terms in the asymptotic expansion of y
corresponding to the characteristic roots with a non-negative real part, and, thus,
construct non-oscillatory m-minor solutions algebraically tending to zero. Further
utilization of this arguments enables us to obtain even more detailed results:
(i) The value a ≤ −((2 − α)π/(2π))α
for α < 2 or a < 0 for α > 2 guarantees
that the rightmost root s1 has a non-negative real part and non-zero imaginary
part (see Propositions 3.1 and 3.2), therefore the major solutions are oscillatory.
(ii)–(iv) If a > 0, Proposition 3.1(i) implies that the rightmost root s1 is a positive
real, therefore the major solutions are non-oscillatory. Eliminating the rightmost
root s1 as in Remark 5.3, the terms corresponding to s2 become dominant and,
again using Proposition 3.2, we obtain the parts (ii)–(iv).
Remark 5.5. For a = 0, (5.2) has the only root s1 = 0 with multiplicity α
and the qualitative behaviour is implied directly by Lemma 2.6(i). In particular, if
α < 1, then all non-trivial solutions of (4.4) are constant, i.e. they are bounded and
non-oscillatory. If α > 1, then all non-trivial solutions of (4.4) are non-oscillatory.
Moreover, if φj = 0 for all j = 1, . . . , α − 1, then the solutions are bounded,
otherwise being unbounded.
14 J. ˇCERM´AK, T. KISELA EJDE-2019/33
It is of a particular interest to emphasize that unlike the integer-order case,
there is no combination of entry parameters such that all the solutions of (4.4) are
oscillatory. In fact, (4.4) has always infinitely many non-oscillatory solutions.
6. Concluding remarks
We have discussed oscillatory and related asymptotic properties of solutions
of the fractional delay differential system (1.6) as well as of the corresponding
scalar equation (4.4). The obtained oscillation results qualitatively differ from
those known from the classical oscillation theory of (integer-order) delay differential
equations. We survey here the most important notes related to this phenomenon.
First, while the appropriate criteria from the classical theory (such as Theorem
1.1) formulate necessary and sufficient conditions for oscillation of all solutions,
their fractional counterparts (Theorem 4.1) present conditions for non-oscillation
of all non-trivial solutions. In particular, our analysis shows that (1.6) cannot admit
only oscillatory solutions. Secondly, considering (1.6), one can observe a close
resemblance between non-oscillation of all non-trivial solutions and convergence to
zero of all solutions (this property defines asymptotic stability of the zero solution
of (1.6)). The latter property is sufficient for non-oscillation of all non-trivial solutions
of (1.6) and, moreover, it is not far from being also a necessary one. These
features (along with some other precisions made in Section 5) demonstrate that
(non)oscillatory properties of (1.6) qualitatively depend on the fact if the value α
is integer or non-integer. In particular, Corollary 4.2 implies that the endpoints
of corresponding non-oscillation intervals depend continuously on changing noninteger
derivative order α; when α is crossing the integer-order value, a sudden
change in oscillatory behaviour occurs (see Corollary 1.2). Note that despite of
some introductory papers on oscillation of (1.6) and other related fractional delay
differential equations (see, e.g. [1, 17]), these properties have not been reported yet.
On the other hand, one can observe that dependence of stability areas of (1.6) on
changing derivative order is “continuous”. As illustrated via Figures 1–4, this area is
continuously becoming smaller, starting from the circle (corresponding to the nondifferential
case when α = 0) to the empty set (when α = 2). We add that the way
to stability remains closed for all real α ≥ 2. From this viewpoint, considerations
of (1.6) with non-integer derivative order enable a better understanding of classical
stability results on (1.6) with integer α.
The method utilized in our oscillation analysis indicates that the main reason of
a rather strange oscillatory behaviour of (1.6) with non-integer α is hidden in the
algebraic rate of convergence of its solutions to zero (compared to the exponential
rate known from the integer-order case). Since this type of convergence has been
earlier described not only for other types of fractional delay equations (see [2, 9,
11, 12]), but also for fractional equations without delay (see [4, 13, 14, 16]), the
above described oscillatory behaviour might be typical for a more general class of
fractional differential equations.
Acknowledgements. This research was supported by the grant 17-03224S from
the Czech Science Foundation.
References
[1] Y. Bolat; On the oscillation of fractional-order delay differential equations with constant
coefficients, Commun. Nonlinear Sci. Numer. Simul., 19 (2014), 3988–3993.
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[2] J. ˇCerm´ak, Z. Doˇsl´a, T. Kisela; Fractional differential equations with a constant delay: Stability
and asymptotics of solutions, Appl. Math. Comput., 298 (2017), 336–350.
[3] J. ˇCerm´ak, J. Horn´ıˇcek, T. Kisela; Stability regions for fractional differential systems with a
time delay, Commun. Nonlinear Sci. Numer. Simulat., 31 (2016), 108–123.
[4] J. ˇCerm´ak, T. Kisela; Stability properties of two-term fractional differential equations, Nonlinear
Dyn., 80 (2015), 1673–1684.
[5] Y. Chen, K.L. Moore; Analytical stability bound for a class of delayed fractional-order dynamic
systems, Nonlinear Dyn., 29 (2012), 191–200.
[6] H.I. Freedman, Y. Kuang; Stability switches in linear scalar neutral delay equations, Funkcial.
Ekvac., 34 (1991), 187–209.
[7] I. Gy˝ori, G. Ladas; Oscillation Theory of Delay Differential Equations: With Applications,
Oxford University Press, Oxford, 1991.
[8] T. Hara, J. Sugie; Stability region for systems of differential-difference equations, Funkcial.
Ekvac., 39 (1996), 69–86.
[9] E. Kaslik, S. Sivasundaram; Analytical and numerical methods for the stability analysis of
linear fractional delay differential equations, J. Comput. Appl. Math., 236 (2012), 4027–4041.
[10] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo; Theory and Applications of Fractional Differential
Equations, Elsevier, Amsterdam, 2006.
[11] K. Krol; Asymptotic properties of fractional delay differential equations, Appl. Math. Comput.,
218 (2011), 1515–1532.
[12] M. Lazarevi´c; Stability and stabilization of fractional order time delay systems, Scientific
Technical Review, 61(1) (2011), 31–44.
[13] C. P. Li, F. R. Zhang; A survey on the stability of fractional differential equations, Eur. Phys.
J. Special Topics, 193 (2011), 27–47.
[14] D. Matignon; Stability results on fractional differential equations with applications to control
processing, IMACS-SMC: Proceedings (1996), 963–968.
[15] I. Podlubn´y; Fractional Differential Equations, Academic Press, San Diego, 1999.
[16] D. Qian, C. Li, R. P. Agarwal, P. J. Y. Wong; Stability analysis of fractional differential
system with Riemann-Liouville derivative, Math. Comput. Modelling 52 (2010), 862-874.
[17] Y. Wang, Z. Han, S. Sun; Comment on “On the oscillation of fractional-order delay differential
equations with constant coefficients” [Commun Nonlinear Sci 19(11) (2014) 3988-3993],
Commun. Nonlinear Sci. Numer. Simulat., 26 (2015), 195–200.
Jan ˇCerm´ak
Institute of Mathematics, Brno University of Technology, Technick´a 2, 616 69 Brno,
Czech Republic
E-mail address: cermak.j@fme.vutbr.cz
Tom´aˇs Kisela
Institute of Mathematics, Brno University of Technology, Technick´a 2, 616 69 Brno,
Czech Republic
E-mail address: kisela@fme.vutbr.cz
Appendix C
Paper on overview for one-term
FDDS [37] (Math Appl, 2020)
Our work on [6,10] brought us significant insights into the stability and asymptotics
of one-term FDDS, but not all findings aligned with the concepts of previous papers.
Consequently, [37] (my author’s share 100 %) aimed to consolidate the topic and to
extend it.
In this paper, we provided a comprehensive overview of the stability and asymptotics
theory for one-term FDDS, considering the two most common definitions
of fractional derivatives: Caputo and Riemann-Liouville. Building on techniques
adopted in our prior research, we derived optimal stability conditions based on the
position of eigenvalues in the complex plane. We elaborated on the implications
of using different definitions of fractional derivative, detailing distinctions on the
stability boundary and in overall asymptotic behaviour.
– 81 –
Math. Appl. 9 (2020), 31–42
DOI: 10.13164/ma.2020.03
ON STABILITY OF DELAYED DIFFERENTIAL SYSTEMS OF
ARBITRARY NON-INTEGER ORDER
TOMÁŠ KISELA
Abstract. This paper summarizes and extends known results on qualitative behavior
of solutions of autonomous fractional differential systems with a time delay. It
utilizes two most common definitions of fractional derivative, Riemann–Liouville
and Caputo one, for which optimal stability conditions are formulated via position
of eigenvalues in the complex plane. Our approach covers differential systems of
any non-integer orders of the derivative. The differences in stability and asymptotic
properties of solutions induced by the type of derivative are pointed out as well.
1. Introduction
In many areas of science and technology we often meet problems which are well
described by differential systems with a time delay. Examples of such situations
might be reaction time of technical and chemical systems or heredity in population
dynamics. Qualitative theory for these equations is summarized in, e.g. [2,5]. The
study of delayed systems involving viscoelasticity, anomalous diffusion or control
theory naturally suggests to enrich our models with derivatives of non-integer order
which proved to be very effective in these areas (see, e.g. [4,8]).
This is the main motivation for our study of two delayed systems which can be
written as
Dα
0 y(t) = Ay(t − τ) , t ∈ (0, ∞), α ∈ R+
\ Z, (1.1)
y(t) = φ(t) , t ∈ [−τ, 0], (1.2)
Dα−k
0 y(t) t=0
= yα−k , k = 1, . . . , α (1.3)
and
C
D
α
0 y(t) = Ay(t − τ) , t ∈ (0, ∞), α ∈ R+
\ Z, (1.4)
y(t) = φ(t) , t ∈ [−τ, 0], (1.5)
y( α −k)
(0) = y α −k , k = 1, . . . , α , (1.6)
where Dα
0 and C
D
α
0 denote the so-called Riemann–Liouville and Caputo differential
operators of order α, respectively. Further, A ∈ Rd×d
is a constant d × d matrix,
y· ∈ Rd
are constant vectors and τ > 0 is a constant delay. As usual for delayed
MSC (2010): primary 34K37, 34K20; secondary 34K25.
Keywords: fractional delay differential system, stability, asymptotic behavior, Riemann–
Liouville derivative, Caputo derivative.
The research was supported by the grant GA20-11846S of the Czech Science Foundation.
31
32 T. KISELA
equations, the initial condition is given by φ ∈ L1
[−τ, 0] (componentwise) and
the use of fractional derivatives allows us to prescribe also initial values for t = 0
separately. We intentionally leave out the integer-order values of α since they
coincide with the known classical cases.
A serious qualitative analysis of such equations is being performed less than
two decades. It spans across scalar and vector cases, various methods like Ddecomposition
or Laplace transform are used. For more details we refer to [1,3,6,9]
which are the main sources for this paper.
The paper is organized as follows. In Section 2 we outline some basic preliminary
results useful in our further considerations. Section 3 is devoted to the discussion of
solution representations and their comparison. The main results are concentrated
in Section 4 where we summarize known facts as well as derive some original ones.
Section 5 concludes the paper with a few final remarks.
2. Preliminaries
Let f be a real function. We use the standard definition of fractional integral of
order γ > 0
Iγ
0 f(t) =
t
0
(t − ξ)γ−1
Γ(γ)
f(ξ)dξ, t ≥ 0 .
We employ both the wide used definitions of fractional derivative of order α > 0
called the Riemann-Liouville and Caputo derivative introduced as
Dα
0 f(t) =
d α
dt α
I
α −α
0 f(t) , t ≥ 0 ,
C
D
α
0 f(t) = I
α −α
0
d α
dt α
f(t) , t ≥ 0 ,
respectively. Additionally, we put C
D
0
0f(t) = D0
0f(t) = f(t) (for more information
on fractional operators we refer, e.g. to [4,8]).
The key tool, utilized throughout this paper, is the Laplace transform which is,
for f, introduced as
L(f(t))(s) =
∞
0
exp{−st}f(t)dt, s ∈ C
provided the integral converges. To perform the transform of (1.1) and (1.4),
we need a clear view on Laplace transform of a function with shifted (delayed)
argument which is given by
L(f(t − τ)h(t − τ))(s) = exp{−τs}L(f(t))(s), τ > 0,
L(f(t − τ))(s) = exp{−τs}L(f(t))(s) + exp{−τs}
0
−τ
exp{−st}f(t)dt, τ > 0 .
(2.1)
STABILITY OF DELAYED FRACTIONAL DIFFERENTIAL SYSTEMS 33
Also, using the formulae for Laplace transform of convolution and power function
L
t
0
f(t − ξ)g(ξ)dξ (s) = L(f(t))(s) · L(g(t))(s),
L
tη
Γ(η + 1)
(s) = s−η−1
, η > −1,
we can see the origin of Laplace transforms of fractional operators
L(Iγ
0 f(t))(s) = s−γ
L(f(t))(s), γ > 0,
L(Dα
0 f(t))(s) = sα
L(f(t))(s) −
α
k=1
sk−1
Dα−k
0 f(t) t=0
, α > 0, (2.2)
L(C
D
α
0 f(t))(s) = sα
L(f(t))(s) −
α
k=1
sα−k
f(k−1)
(0), α > 0. (2.3)
The symbol h denotes the Heaviside step function defined as h(ξ) = 1 for ξ ≥ 0
and h(ξ) = 0 for ξ < 0. When applied on a vector function, the Laplace transform
is considered componentwise.
We note that the system matrix A of (1.1) and (1.4) can be rewritten with
the use of a matrix Λ in a Jordan canonical form with the Jordan blocks on its
diagonal as A = TΛT−1
, where T is a regular real d × d matrix,
Λ =





J1 0 · · · 0
0 J2 · · · 0
...
...
...
...
0 0 · · · Jq





, Jk =








λi 1 0 · · · 0
0 λi 1
...
...
...
...
...
... 0
0 · · · λi 1
0 · · · 0 λi








, k = 1, . . . , q,
and λi (i = 1, . . . , n) are distinct eigenvalues of A. The number of Jordan blocks
corresponding to λi is called geometric multiplicity of λi. The sum of the sizes of
all Jordan blocks corresponding to λi is called algebraic multiplicity of λi.
Before we proceed to the next section, we recall the stability notions related to
our linear fractional differential systems with a delay. The zero solution is said to
be stable (asymptotically stable) if the solution of the system is bounded (tends
to zero as t → ∞) for any initial function φ ∈ L1
([−τ, 0]).
3. Solution representations for (1.1) and (1.4)
As in the integer-order case (see, e.g. [2,5]), an essential role is played by analogue
of the fundamental matrix solution also for (1.1) and (1.4) (see, e.g. [1]). In order
to simplify the notation dealing with the orders α greater than one, we introduce
its generalization in form of the following functions
RA,τ
α,β (t) = L−1
(sα
I − A exp{−sτ})−1
sα−β
(t) , α ∈ R+
\ Z, β ∈ R+
where A ∈ Rd×d
and I is the identity d × d matrix. Employing these R-functions,
we arrive at the following solution representations.
34 T. KISELA
Theorem 3.1. The solution yRL of (1.1)–(1.3) is given by
yRL(t) =
α
k=1
RA,τ
α,α−k+1(t)yα−k +
0
−τ
RA,τ
α,α(t − τ − u)Aφ(u)du.
Proof. Applying (2.1), (2.2) on (1.1)–(1.3), we get
L(y(t))(s)
= (sα
I − A exp{−sτ})−1


α
k=1
sk−1
yα−k +
0
−τ
exp{−s(t + τ)}Aφ(t)dt


=
α
k=1
L(RA,τ
α,α−k+1(t))(s)yα−k +
0
−τ
exp{−s(t + τ)}L(RA,τ
α,α(t))(s)Aφ(t)dt
which yields the assertion.
Theorem 3.2. The solution yC of (1.4)–(1.6) is given by
yC(t) =
α
k=1
RA,τ
α,k (t)yk−1 +
0
−τ
RA,τ
α,α(t − τ − u)Aφ(u)du.
Proof. Analogously as above, applying (2.1), (2.3) on (1.4)–(1.6), we obtain
L(y(t))(s)
= (sα
I − A exp{−sτ})−1


α
k=1
sα−k
yk−1 +
0
−τ
exp{−s(t + τ)}Aφ(t)dt


=
α
k=1
L(RA,τ
α,k (t))(s)yk−1 +
0
−τ
exp{−s(t + τ)}L(RA,τ
α,α(t))(s)Aφ(t)dt
which again concludes the proof.
Remark 3.3. We can see that the integral terms involving the initial function φ
are for yRL and yC identical. The difference occurs in the terms involving the local
initial conditions. Although the Caputo case is more studied in the literature, in
particular of order α ∈ (0, 1] (see, e.g. [1,3,6]), the Riemann-Liouville one actually
appears to be structurally closer to the classical case. Indeed, RA,τ
α,α seems to be
playing practically the same role as the fundamental matrix solution in integerorder
delay differential equations.
It might look like Theorems 3.1 and 3.2 are not that much explicit since the
R-functions are defined via the inverse Laplace transform. Now we show that these
functions can be actually evaluated pretty straighforwardly.
Applying the Jordan canonical form theory, we can write
L(RA,τ
α,β (t))(s) = (sα
I − A exp{−sτ})−1
sα−β
= T(sα
I − Λ exp{−sτ})−1
sα−β
T−1
.
STABILITY OF DELAYED FRACTIONAL DIFFERENTIAL SYSTEMS 35
Clearly, the matrix (sα
I − Λ exp{−sτ})−1
sα−β
is block diagonal with the blocks
given by upper triangular strip matrices of the form
(sα
I − Jke−sτ
)−1
sα−β
=







sα−β
sα−λie−sτ
e−sτ
sα−β
(sα−λie−sτ )2 · · · e−(r−1)sτ
sα−β
(sα−λie−sτ )rk
0 sα−β
sα−λie−sτ
... e−(r−2)sτ
sα−β
(sα−λie−sτ )rk−1
...
...
...
...
0 0 · · · sα−β
sα−λie−sτ







,
(3.1)
where Jk (k = 1, . . . , q) is the k-th block of Λ and rk is its size. It was proven in
[1] that the elements of this matrix can be expressed as
exp{−msτ}sα−β
(sα − λ exp{−sτ})m+1
= L(Gλ,τ,m
α,β (t))(s)
where
Gλ,τ,m
α,β (t) =
t/τ−m−1
j=0
m + j
j
λj
(t − (m + j)τ)α(m+j)+β−1
Γ(α(m + j) + β)
, t > 0 .
To summarize the previous considerations, we can write the following assertion.
Lemma 3.4. Let A ∈ Rd×d
, λi (i = 1, . . . , n) be distinct eigenvalues of A and
let pi be the largest size of the Jordan block corresponding to the eigenvalue λi.
Then the non-zero elements of matrix function RA,τ
α,β are linear combinations of
scalar functions
Gλi,τ,m
α,β (t), m = 0, . . . , pi − 1, i = 1, . . . , n .
4. Main results
It is well known from the basic theory of the Laplace transform method that if
all poles of the Laplace image of solutions (roots of the so-called characteristic
equation) have negative real parts, then the zero solution of the studied equation
is asymptotically stable (and their non-zero solutions tend to zero in an exponential
rate). On the other hand, if there exists a pole with a positive real part,
the corresponding zero solution is not stable (its absolute value tends to infinity
exponentially). In the fractional case, it usually occurs a more complex situation,
involving also singular points and poles with the zero real parts, which require
a deeper analysis.
For our fractional problems (1.1) and (1.4), as it can be seen from the proof of
Theorems 3.1 and 3.2, the characteristic equation takes the form
det(sα
I − A exp{−sτ}) = 0 or
n
i=1
(sα
− λi exp{−sτ})wi
= 0 , (4.1)
where λi (i = 1, . . . , n) are distinct eigenvalues of A and wi are the corresponding
algebraic multiplicities. As we can see from (4.1) and (3.1), for further eigenvalues
considerations it is sufficient to investigate the roots of the equation
p(s; λ) ≡ sα
− λ exp{−sτ} = 0 (4.2)
36 T. KISELA
where λ is a complex parameter. Now, we perform a direct root analysis of (4.2).
In particular, we formulate the optimal conditions on λ ensuring that (4.2) does
not have any root with positive real part.
Lemma 4.1. Let α ∈ R+
\ Z, τ > 0 and λ ∈ C. Then all the roots of (4.2)
have negative real parts if and only if
α ∈ (0, 2) , |Arg (λ)| >
απ
2
and |λ| <
|Arg (λ)| − απ/2
τ
α
(4.3)
where Arg (λ) ∈ (−π, π] is the principal argument of λ.
Proof. The case λ = 0 is trivial since then (4.2) has only the zero solution which
does not satisfy (4.3). Let λ = 0 and put
s = r exp{iϕ} , λ = exp{iψ}
where r = |s|, = |λ| and ϕ, ψ ∈ (−π, π] are principal arguments of s, λ, respectively.
Then we can write (4.2) for real and imaginary parts as a system of two
equations in the form
rα
cos(αϕ) − exp{−rτ cos(ϕ)} cos(ψ − rτ sin(ϕ)) = 0, (4.4)
rα
sin(αϕ) − exp{−rτ cos(ϕ)} sin(ψ − rτ sin(ϕ)) = 0. (4.5)
Now, let us assume that (4.2) has a root with a non-negative real part, i.e.
|ϕ| ≤ π/2.
For ϕ = 0, we have ψ = 0 (i.e. λ = ) from (4.5). Further, (4.4) implies, for r
and , the relation rα
= exp{−rτ} which always allows to find an appropriate r
to a given . Hence, (4.2) has a non-negative real root if and only if λ is a nonnegative
real.
Let |ϕ| ∈ (0, π/2] \ {π/α}. Since |ϕ| = π/α, we have ψ − rτ sin(ϕ) = kπ for
any k ∈ Z and, by dividing and rearranging (4.4) and (4.5), we arrive at a new
reformulation of (4.4), (4.5) in the form
αϕ = ψ − rτ sin(ϕ) + 2kπ, (4.6)
rα
= exp{−rτ cos(ϕ)} (4.7)
for a suitable k ∈ Z (the replacement of kπ by 2kπ is implied by positivity of r
and ). Further, by eliminating r from (4.6), (4.7), we get the equation for ϕ as
ψ − αϕ + 2kπ
τ sin(ϕ)
α
= exp{(αϕ − ψ − 2kπ) cot(ϕ)} .
As proven in [1] for α ∈ (0, 1), the left-hand side is decreasing with respect to ϕ
on (0, π/2] with the lowest value at ϕ = π/2 for any k. The right-hand side is
increasing with respect to ϕ on (0, π/2] with the largest value at ϕ = π/2 for any
k. It can be easily checked that the situation for α ≥ 1 is the same provided we
put the left-hand side equal to zero for ϕ such that ψ − αϕ + 2kπ < 0. Obviously,
the existence of a root ϕ ∈ (0, π/2] for at least one k is ensured if and only if
|ψ| ≤
απ
2
or ≥
|ψ| − απ/2
τ
α
. (4.8)
STABILITY OF DELAYED FRACTIONAL DIFFERENTIAL SYSTEMS 37
We can see that (4.8)1 is automatically satisfied for α ≥ 2, hence for α ≥ 2 there is
always a root of (4.2) with a non-negative real part. We can see that, for α ∈ (0, 2),
(4.8) is a complement of (4.3).
So far, we have not investigated the situation |ϕ| = π/α ≤ π/2. However, it
can occur only for α ≥ 2 and in that case we already know that there is always
a root of (4.2) with a non-negative real part.
Summarizing the previous arguments, we can conclude the proof.
Lemma 3.4 shows that functions of the type Gλ,τ,m
α,β play, for (1.1) and (1.4),
an analogous role as exponential functions for integer-order systems. Hence, it
is crucial to have a good uderstanding of asymptotic behavior of Gλ,τ,m
α,β and its
relation to (4.2) which is provided by the following assertion which slightly extends
the result presented in [1].
Lemma 4.2. Let λ ∈ C, α, β, τ ∈ R+
and m ∈ Z be such that α ∈ R+
\ Z,
m ≥ 0. Further, let si (i = 1, 2, . . . ) be the roots of (4.2) with ordering (si) ≥
(si+1) (in particular, s1 is the zero with the largest real part).
(i) If λ = 0, then
G0,τ,m
α,β (t) =
(t − mτ)mα+β−1
Γ(mα + β)
h(t − mτ).
(ii) If λ is such that s1 has negative real part, then
Gλ,τ,m
α,β (t) =
(−1)m+1
λm+1Γ(β − α)
(t + τ)β−α−1
+
(−1)m+1
(m + 1)
λm+2Γ(β − 2α)
(t + 2τ)β−2α−1
+ O(tβ−3α−1
) as t → ∞.
(iii) If λ is such that s1 is purely imaginary or it has positive real part, then
Gλ,τ,m
α,β (t) =
m
j=0
(t − mτ)j
(aj exp{s1(t − mτ)}
+ bj exp{s2(t − mτ)}) +
O(tm
exp{ (s3)t}), if (s3) ≥ 0,
O(tβ−α−1
), if (s3) < 0
as t → ∞
where aj, bj are suitable nonzero complex constants (j = 0, . . . , m).
Proof. The assertion was proved in [1] for the case α ∈ (0, 1). The generalization
for α > 1 is a tedious but direct analogue.
Now we are in a position to formulate the main results of this paper. For the
sake of lucidity, we introduce the following subset of complex numbers motivated
by (4.3) as
Sα,τ = λ ∈ C : |λ| <
|Arg (λ)| − απ/2
τ
α
, |Arg (λ)| >
απ
2
,
which we call the stability region of (1.1) and (1.4).
38 T. KISELA
Theorem 4.3. Let A ∈ Rd×d
, α ∈ R+
\ Z and τ > 0. Further, let p0 ∈ Z
be the largest size of the Jordan block corresponding to the zero eigenvalue of A,
where we put p0 = 0 if A has only non-zero eigenvalues.
(i) The zero solution of (1.1) is asymptotically stable if and only if α ∈ (0, 2),
all non-zero eigenvalues of A belong to Sα,τ and p0 < 1/α.
(ii) The zero solution of (1.1) is stable if and only if α ∈ (0, 2), all eigenvalues
of A belong to cl (Sα,τ ), all non-zero eigenvalues of A lying on ∂Sα,τ have
the same algebraic and geometric multiplicities and p0 ≤ 1/α.
Proof. Theorem 3.1 and Lemma 3.4 imply that the solution components of (1.1)
are formed as linear combinations of functions
Gλi,τ,m
α,α−k+1(t) and
0
−τ
Gλi,τ,m
α,α (t − τ − u)φj(u)du , (4.9)
where k = 1, . . . , α , λi (i = 1, . . . , n) are eigenvalues of A, m is a non-negative
integer as specified in Lemma 3.4 and φj (j = 1, . . . , d) are components of the
initial function.
Lemma 4.1 implies that all roots of (4.1) have negative real part if and only if
α ∈ (0, 2) and all eigenvalues belong to Sα,τ . Moreover, (4.1) has at least one root
with zero real part and other roots with a negative real part if and only if at least
one eigenvalue lies on the boundary of Sα,τ .
Thus, the asymptotic behavior of the solution can be derived from Lemma 4.2.
The functions (4.9)1 are described directly, we just point out that for λi ∈ Sα,τ
the first term in the expansion cancels out due to the negative integer argument
in the Gamma function, so that we obtain
Gλi,τ,m
α,α−k+1(t) =
(−1)m+1
(m + 1)
λm+2
i Γ(−α − k + 1)
(t + 2τ)−α−k
+ O(t−2α−k
) as t → ∞.
Now, we investigate (4.9)2. Employing the assuption φ ∈ L1
[−τ, 0] and Lemma
4.2, we can distinguish several cases:
Let α ∈ (0, 2) and λi ∈ Sα,τ . The second mean value theorem implies
0
−τ
Gλi,τ,m
α,α (t − τ − u)φ(u)du = Gλi,τ,m
α,α (t)
ξ
−τ
φ(u)du
= K1(t + 2τ)−α−1
+ O(t−2α−1
) as t → ∞ ,
where K1 ∈ R is non-zero and ξ ∈ (−τ, 0].
Now, let λi = 0. By the same approach we arrive at
0
−τ
G0,τ,m
α,α (t − τ − u)φ(u)du = K2(t − mτ)(m+1)α−1
,
where K2 ∈ R is non-zero. This expression vanishes for t → ∞, if and only if
m + 1 = p0 < 1/α.
The cases for λi ∈ ∂Sα,τ \ {0} and λi /∈ cl (Sα,τ ) can be handled similarly.
We arrive at the conclusion that (4.9)2 is bounded, when the non-zero eigenvalue
lying on the boundary of stability region has the same algebraic and geometric
multiplicity. Otherwise the absolute value of (4.9)2 increases polynomially (when
the eigenvalue lies on the boundary) or exponentially.
STABILITY OF DELAYED FRACTIONAL DIFFERENTIAL SYSTEMS 39
Theorem 4.4. Let A ∈ Rd×d
, α ∈ R+
\ Z and τ > 0. Further, let p0 ∈ Z
be the largest size of the Jordan block corresponding to the zero eigenvalue of A,
where we put p0 = 0 if A has only non-zero eigenvalues.
(i) The zero solution of (1.4) is asymptotically stable if and only if α ∈ (0, 2)
and all eigenvalues of A belong to Sα,τ .
(ii) The zero solution of (1.4) is stable if and only if α ∈ (0, 2], all eigenvalues
of A belong to cl (Sα,τ ), all non-zero eigenvalues of A lying on ∂Sα,τ have
the same algebraic and geometric multiplicities and p0 ≤ 2 − α .
Proof. The idea of the proof is equivalent to that one of Theorem 4.3. In
particular, the solution components of (1.4) are given by linear combinations of
Gλi,τ,m
α,k (t) and
0
−τ
Gλi,τ,m
α,α (t − τ − u)φj(u)du , (4.10)
where k = 1, . . . , α , λi (i = 1, . . . , n) are eigenvalues of A, m is a non-negative
integer as specified in Lemma 3.4 and φj (j = 1, . . . , d) is a component of the initial
function. Thus, we see that (4.10)2 is the same as (4.9)2 while (4.10)1 differs with
respect to (4.9)1 due to the change of index. This causes only a different decay
rate for λi ∈ cl (Sα,τ ).
Overall, there is only one difference in stability behavior which occurs for λi = 0
when we have
G0,τ,m
α,k (t) =
(t − mτ)mα+k−1
Γ(mα + k)
.
We can see that this function never tends to zero with t → ∞ and it is bounded
if and only if mα + k − 1 = 0 which means α = 1 (i.e. k = 1) and p0 = 1 (i.e.
m = 0).
Remark 4.5. (i) Theorems 4.3 and 4.4 show that Sα,τ is the stability region for
delayed fractional differential systems for Riemann-Liouville and Caputo derivative,
i.e. for (1.1) and (1.4), respectively. Figure 1 represents the situation for
α ∈ (0, 1) when the stability region includes also points with positive real part.
We can see in Figure 2 how the region is transformed for α ∈ (1, 2), and it is apparent
how the stability region vanishes for α → 2. Also, for τ → 0, Sα,τ tends to
the stability region known from theory of fractional differential equations without
delay (see, e.g. [7,9]).
(ii) From the stability viewpoint, the only difference between (1.1) and (1.4)
occurs if there is a zero eigenvalue and the order of derivatives is less than 1. In
this case, the zero solution to (1.1) can be asymptotically stable, stable or unstable,
depending on the particular value of α and multiplicities of the zero eigenvalue.
The zero solution of (1.4) is stable if algebraic and geometric multiplicities of the
zero eigenvalue are equal, otherwise it is unstable (i.e. it does not depend on the
particular value of α).
The proof technique used for Theorems 4.3 and 4.4 actually reveals more than
the stability properties. Due to its constructive nature we can actually derive also
the asymptotic behavior of the solutions to (1.1) and (1.4). We summarize the
comparisons of the two cases in the following assertions dealing with the asymptotic
40 T. KISELA
𝛼
)
𝛼 𝛼𝜋
2
ℜ(𝜆)
ℑ(𝜆)
𝑆 𝛼,𝜏)
))
𝛼𝜋
2
𝜋 − 𝛼𝜋/2
𝜏
𝜋/2 − 𝛼𝜋/2
𝜏
Figure 1. The stability region Sα,τ for the values α = 0.4 and τ = 1.
ℜ(𝜆)
ℑ(𝜆)
)
𝛼
) 𝜋 − 𝛼𝜋/2
𝜏
𝑆 𝛼,𝜏
𝛼𝜋
2
𝛼𝜋
2
Figure 2. The stability region Sα,τ for the values α = 1.1 and τ = 1.
STABILITY OF DELAYED FRACTIONAL DIFFERENTIAL SYSTEMS 41
equivalence (denoted by the symbol ∼) relationships for norms of solutions (we
use the symbol · for Euclidean norm in Rd
).
Theorem 4.6. Let A ∈ Rd×d
, α ∈ (0, 2), τ > 0 and let all the eigenvalues of
A belong to Sα,τ . Further, we denote by yRL and yC the solutions of (1.1)–(1.3)
and (1.4)–(1.6), respectively. Then it holds
yRL(t) ∼ t−α−1
and yC(t) ∼ t α −α−1
as t → ∞ (4.11)
for almost all choices of initial conditions. If yRL and yC do not follow (4.11),
then their norms tend to zero with a faster decay rate.
Proof. Theorems 3.1 and 3.2 indicate some particular choices of initial conditions,
e.g. y0 = 0, which can remove the dominating terms from yRL and yC
and therefore affect the decay rate. The particular asymptotic properties are then
implied by Lemma 4.2.
Theorem 4.7. Let A ∈ Rd×d
, α ∈ (0, 2) and τ > 0. Let A has the zero
eigenvalue and denote p0 the size of the largest Jordan block corresponding to
this zero eigenvalue. Let all non-zero eigenvalues of A belong to Sα,τ . Further,
we denote yRL and yC the solutions of (1.1)–(1.3) and (1.4)–(1.6), respectively.
Then it holds
yRL(t) ∼ tp0α−1
and yC(t) ∼ t(p0−1)α+ α −1
as t → ∞ (4.12)
for almost all choices of initial conditions. If yRL and yC do not follow (4.12),
then their norms are even smaller for t large enough.
Proof. The idea of the proof is analogous to the previous case.
Remark 4.8. (i) We can observe an interesting distinction between the way
how the asymptotic behavior of yRL and yC depends on α. While in the Riemann–
Liouville case we see the algebraic decay rate depending directly on α, in the
Caputo case the decay rate is driven by the decimal part of α, i.e. by the difference
α − α. Indeed, if we consider for example α1 = 0.4 and α2 = 1.4, then
the solutions of (1.4) follow essentially the same asymptotic relations, while the
Riemann–Liouville ones do not.
(ii) We can employ a similar analysis also in the cases that are not covered by
Theorems 4.6 and 4.7, i.e. when there is a non-zero eigenvalue on the boundary
or outside the closure of the stability region. We note that if there is a non-zero
eigenvalue lying outside the closure of the stability region, the norms of non-zero
solutions increase exponentially for both (1.1) and (1.4).
(iii) We point out that the asymptotic results obtained for the delayed fractional
differential systems actually mirror the well-known results for fractional differential
systems without a delay.
5. Conclusions
We have summarized and extended the results on qualitative behavior of solutions
of delayed fractional differential systems (1.1) and (1.4) of arbitrary order.
We have shown that the stability of the zero solution occurs only if the order
of derivatives is less than 2. Further, we have derived the precise description of
42 T. KISELA
the stability region which is for both (1.1) and (1.4) identical. The only difference
regarding the stability occurs when the system matrix A has a zero eigenvalue.
Then we observe the asymptotic stability property for (1.1) only if α < 1 and
the maximum size of the Jordan block corresponding to the zero eigenvalue being
less than 1/α. In the Caputo case (1.4), the asymptotic stability does not appear
and the zero solution is stable (but not asymptotically stable) only if α < 1 and
algebraic multiplicity of the zero eigenvalue being equal to the geometric one.
The asymptotic behavior displays more diversity. If the system matrix A has
all eigenvalues lying in the stability region, i.e. the zero solutions of both (1.1) and
(1.4) are asymptotically stable, we can generally say that the solutions of (1.1) go
to zero as t → ∞ faster that solutions of (1.4). Moreover, unlike the RiemannLiouville
case, the decay rate of solutions to (1.4) does not depend on the value α
itself, but on its decimal part only.
The area of qualitative analysis of fractional differential equations with a time
delay, especially with higher-order derivative, provides a lot of open problems. Our
research may serve as one of the prerequisites to studies of more complex systems,
such as Dα
0 y(t) = ay(t) + by(t − τ) or its vector analogues.
References
[1] J. Čermák, J. Horníček and T. Kisela, Stability regions for fractional differential systems
with a time delay, Communications in Nonlinear Science and Numerical Simulation 31
(2016), 108–123.
[2] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations,
Springer-Verlag, New York, 1993.
[3] E. Kaslik and S. Sivasundaram, Analytical and numerical methods for the stability analysis
of linear fractional delay differential equations, Journal of Computational and Applied
Mathematics 236 (2012), 4027–4041.
[4] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional
Differential Equations, Elsevier, Amsterdam, 2006.
[5] V. Kolmanovskii and A. Myshkis, Introduction to the Theory and Applications of Functional
Differential Equations, Kluwer Academic Publishers, Dordrecht, 1999.
[6] K. Krol, Asymptotic properties of fractional delay differential equations, Applied Mathematics
and Computation 218 (2011), 1515–1532.
[7] D. Matignon, Stability results on fractional differential equations with applications to control
processing, in: Computational Engineering in Systems Applications, vol. 2, IMACS, IEEE–
SMC, Lille, 1996, pp. 963–968.
[8] I. Podlubný, Fractional Differential Equations, Academic Press, San Diego, 1999.
[9] D. Qian, C. Li, R. P. Agarwal and P. J. Y. Wong, Stability analysis of fractional differential
systems with Riemann–Liouville derivative, Mathematical and Computer Modelling 52
(2010), 862–874.
Tomáš Kisela, Institute of Mathematics, Brno University of Technology, Technická 2, Brno
616 69, Czechia
e-mail: kisela@fme.vutbr.cz
Appendix C. Overview for one-term FDDS (Math Appl, 2020) 94
Appendix D
Paper on Lambert function and
one-term FDDE [15] (FCAA, 2023)
The topic of one-term FDDS seemed, in principle, mostly complete to us, at least
regarding stable solutions. We revisited it only recently in a re-union of the author’s
trio, which was primarily active before 2016. The inspiration for [15] (co-authors: J.
Čermák, L. Nechvátal; my author’s share 33 %) came from discussions nearly nine
years ago about a possible fractional generalization of the classical Lambert function
technique known from the stability analysis of ordinary delay differential equations.
As it turned out, this generalization is not only possible but, in some cases, easier
than other known approaches. We managed to re-derive fully explicit criteria that
had not been reached by this technique before, contributing to a better understanding
of unbounded solutions for higher-order FDDS. As a by-product, we created
an "asymptotic map" for unbounded solutions, showing their large-time exponential
modulus growth and frequency of oscillations based on the location of system
matrix eigenvalue.
– 95 –
Fractional Calculus and Applied Analysis (2023) 26:1545–1565
https://doi.org/10.1007/s13540-023-00176-x
ORIGINAL PAPER
The Lambert function method in qualitative analysis of
fractional delay differential equations
Jan ˇCermák1 · Tomáš Kisela1 · Ludˇek Nechvátal1
Received: 13 December 2022 / Revised: 19 May 2023 / Accepted: 25 May 2023 /
Published online: 16 June 2023
© The Author(s) 2023
Abstract
We discuss an analytical method for qualitative investigations of linear fractional delay
differential equations. This method originates from the Lambert function technique
that is traditionally used in stability analysis of ordinary delay differential equations.
Contrary to the existing results based on such a technique, we show that the method
can result into fully explicit stability criteria for a linear fractional delay differential
equation, supported by a precise description of its asymptotics. As a by-product of
our investigations, we also state alternate proofs of some classical assertions that are
given in a more lucid form compared to the existing proofs.
Keywords Fractional delay differential equation (primary) · Lambert function ·
Stability · Asymptotic behavior
Mathematics Subject Classification (Primary) 34K37 · 33E30 · 33E12 · 34K20 ·
34K25
1 Introduction
The paper discusses an analytical method for qualitative investigations of fractional
delay differential equations (FDDEs). These equations are currently very intensively
studied due to their importance in various application areas, with a special emphasis to
control theory. Indeed, presence of both the time lag as well as non-integer derivative
B Ludˇek Nechvátal
nechvatal@fme.vutbr.cz
Jan ˇCermák
cermak.j@fme.vutbr.cz
Tomáš Kisela
kisela@fme.vutbr.cz
1 Institute of Mathematics, Brno University of Technology, Technická 2896/2, 61669 Brno, Czechia
123
1546 J. ˇCermák et al.
order as control or tunning parameters in studied models provides a very efficient tool
for various control processes such as stabilization or destabilization of the particular
solutions of these models (for a pioneering work in this direction we refer to [17]).
Systematic investigations of FDDEs were initiated in the paper [9]. Here, stability
properties of
Dα
x(t) = λx(t − τ), (1)
where α, τ ∈ R+, λ ∈ R and Dα is a fractional differential operator, were analyzed
using the fact that (1) is asymptotically stable (i.e., any its solution is eventually tending
to zero) if and only if all the roots of the characteristic equation
sα
− λ exp(−sτ) = 0 (2)
have negative real parts. To explore such a location of characteristic roots with respect
to the imaginary axis, the Lambert function technique was utilized. The essence of
the method consists in a representation formula for characteristic roots in terms of
appropriate branches of this multi-valued function (for some precisions concerning
the correct use of the Lambert function technique in stability analysis of (1), we refer
also to [12]). A certain general disadvantage of this approach consists in its (seeming)
disability to provide stability criteria in an explicit form depending on entry parameters
only (i.e., on α, λ and τ in the case of (1)).
As the other papers on stability and asymptotic properties of (1) followed, the
Lambert function method was replaced by some alternate classical tools of stability
investigations (such as D–partition method or τ-decomposition method) modified to
the fractional case. Using these approaches, effective and non-improvable stability
conditions for (1), supported by some asymptotic bounds, were derived in [16] (the
case λ ∈ R, 0 < α < 1), [6] (the case λ ∈ C, 0 < α < 1), and partially also in [7]
(the case λ ∈ C, α > 0). Some of the mentioned results can be extended also to the
case of a two–term FDDE
Dα
x(t) = μx(t) + λx(t − τ). (3)
In this respect, we refer to [2, 5, 15] (the case μ, λ ∈ R, 0 < α < 1) and [8] (the
case μ, λ ∈ R, 1 < α < 2). Following the integer-order case (see, e.g., [1]), (1) and
(3) may serve as test equations for numerical analysis of FDDEs. From this point of
view, it is very important to describe their basic qualitative properties in the strongest
possible form. Then, when analyzing appropriate numerical schemes applied to these
test equations, the ability to keep the key qualitative properties of the underlying exact
equations is of basic importance. For some other recent advances in qualitative theory
of FDDEs, we refer, e.g., to [3, 10, 11, 18–20, 23].
Following the above outlines, the aim of this paper is twofold. First, we deepen the
existing knowledge on some qualitative properties of (1) with the Caputo fractional
derivative. Second, perhaps a more important aspect of the paper consists in the way
how we aim to do it. We come back to the Lambert function method used in [9]
and show that this approach can offer more than formulae depending on the use of
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The Lambert function method in qualitative... 1547
supporting software packages. In fact, this technique can result into actually effective
stability and asymptotic criteria.
Thepaperisorganizedasfollows.Section2recallssomeexistingfindingson (1)and
essentials of the Lambert function theory. In Sect. 3, we explore the Lambert function
method in details. In particular, we give an alternate proof of the classical assertion
saying that the characteristic root generated by the principal branch of the Lambert
function has the largest real part, and formulate a criterion that enables to localize
values of the principal branch in the complex plane. Section 4 presents applications of
these results to (1). Here, we extend the existing stability criteria for (1) to arbitrary
(positive) real values of α, and formulate sharp asymptotic estimates for the solutions
of (1). Some final remarks in Sect. 5 conclude the paper.
2 Basic mathematical background
In this section, we summarize some known facts relevant to our next investigations.
First, we recall a close relationship between stability and asymptotic properties of (1),
and distribution of the characteristic roots of (2). Then, we recall some basics of the
Lambert function and its use in stability analysis of FDDEs.
It was shown in [7] that any solution x of (1) with the Caputo fractional derivative
(and a generally complex λ) can be written using the Mittag-Leffler type function
Gλ,τ
α,β(t) =
t/τ −1
j=0
λj (t − jτ)α j+β−1
(α j + β)
, α, β > 0, (4)
where · denotes the upper integer part. More precisely, if φ is a continuous initial
(complex-valued) function on [−τ, 0], φ0 = φ(0) and φj , j = 1, . . . , α − 1, are
(complex) constants (considered when α > 1), then
x(t) =
α −1
j=0
φj Gλ,τ
α, j+1(t) + λ
0
−τ
Gλ,τ
α,α(t − τ − ξ)φ(ξ) dξ (5)
is the solution of (1) satisfying x(t) = φ(t) for all t ∈ [−τ, 0], and limt→0+ x( j)(t) =
φj , j = 1, . . . , α − 1.
Based on some asymptotic results on (4), the solution (5) can be rewritten by the
use of the characteristic roots having non-negative real parts. We recall that (2) admits
countably many roots, and only a finite number of them is lying right to any line
(s) = p, p ∈ R (throughout the paper, the symbol (z) and (z) stands for the real
and imaginary part of z ∈ C, respectively). If we denote by S the set of all roots of (2)
having non-negative real parts (note that S must be a finite set), then, for a non-integer
α, (5) can be rewritten as
x(t) =
s∈S
cs exp(st) + O(t j−α
) as t → ∞ (6)
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1548 J. ˇCermák et al.
where cs are complex coefficients depending on α, τ, λ, φ, and j ∈ {−1, 0, . . . , α −
1} (the particular value of j depends on limit behavior of φ at t = 0). Notice that
j − α < 0, i.e., the function t j−α always tends to zero.
By (6), the roots of (2) play an essential role in qualitative behavior of the solutions
of (1). Following the classical integer-order pattern, the authors in [9] used the
following chain of steps
sα
exp(sτ) = λ → s exp
τ
α
s = λ
1
α →
τ
α
s exp
τ
α
s =
τ
α
λ
1
α (7)
to express the roots of (2) via the Lambert function introduced as the solution of
W(z) exp(W(z)) = z, z ∈ C. (8)
Before we recall the root formula for (2) based on this special function, some of its
basic properties might be collected. The Lambert function is a multi-valued function
(except at z = 0) with infinitely many (single-valued) branches Wk, k ∈ Z. Neither
of them can be expressed in terms of elementary functions. In particular, W0 is called
a principal branch. For any z ∈ C, (W0(z)) is between −π and π. The other branches
are numbered so that (Wk(z)) is between (2k −2)π and (2k +1)π while (W−k(z))
is between −(2k + 1)π and −(2k − 2)π for any z ∈ C and k = 1, 2, . . .. More
precisely, the ranges of W±k and W±(k+1), k = 0, 1, . . . , are separated by the curves
{w = x + i y ∈ C : x = −y cot(y), 2kπ < |y| < (2k + 1)π}
and the ranges of W1 and W−1 are separated by the half-line
{w = x + i y ∈ C : −∞ < x ≤ −1, y = 0}.
These separating curves correspond to the branch cuts in the z-plane defined as
{z = ξ + i η ∈ C : −∞ < ξ ≤ − exp(−1), η = 0}
in the case of W0, and
{z = ξ + i η ∈ C : −∞ < ξ ≤ 0, η = 0}
in the case of Wk, k = 0. Conventionally, the branch cut (having the argument π
in the z-plane) is mapped by Wk on its upper boundary in the w-plane. Only the
branches W0 and W−1 take on real values for a real z ∈ [− exp(−1), ∞) and a real
z ∈ [− exp(−1), 0), respectively. Further details on the Lambert function (including
some historical remarks) can be found in [4], for other comments, see also [13] and
[22].
Now, following (7), all the roots of (2) can be expressed in the form
sk =
α
τ
Wk
τ
α
λ
1
α , k ∈ Z. (9)
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The Lambert function method in qualitative... 1549
By (6), a crucial role in analysis of (1) is played by the rightmost characteristic root
(i.e., the root of (2) with the largest real part). The following classical assertion says
that this root is just s0.
Lemma 1 Let z ∈ C. Then W0(z) has the largest real part (W0(z)) among all the
other real parts (Wk(z)), k ∈ Z.
The original proof of Lemma 1 is pretty long (see [22]). As a by-product of our
next procedures, we are going to present an alternate (and more simple) way how to
prove this assertion.
Remark 1 As pointed out in [12], the expression (9) is not quite correct for some
complex values of λ. More precisely, (7) contains taking the 1/α-power which means
that the roots given by (9) are identical to those of (2) only in the case
|Arg(λ)| ≤ απ
(we recall that −π < Arg(·) ≤ π). This inequality is satisfied trivially when α ≥ 1
but makes a restriction when 0 < α < 1. In other words, if | Arg(λ)| > απ, then the
representation (9) can produce some superfluous roots that are actually not the true
roots of (2). As an example, we can consider, e.g., the case λ = −1, α = 1/2 and τ = 1
when (2) has the rightmost root s0 ≈ −0.4172 − i 2.2651 (i.e., (1) is asymptotically
stable) while (9) produces s0 ≈ 0.4263 > 0. On this account, we discuss qualitative
properties of (1) for α > 1. Comments to the case 0 < α < 1 are provided in the final
section.
3 Some advances on the Lambert W function
This section contains several key results on the Lambert function which proved to be
useful in qualitative investigations of (1). To obtain an actually effective and strong
asymptotic description of the solutions of (1), we need to effectively localize the
position of the rightmost characteristic root in the complex plane. More precisely,
by (6) and (9), we need to derive effective expressions of the real and imaginary
parts of W0(z) in terms of z. Thus, keeping in mind intended stability and asymptotic
analysis of (1), we can pose the following problems: For given p ∈ R and z ∈ C, is
it possible to characterize the properties (W0(z)) < p and (W0(z)) = p directly
in terms of z and p, i.e., without an evaluation of the principal branch of the Lambert
function? Further, for given q ∈ R and z ∈ C, is it possible to similarly elaborate on
the properties | (W0(z))| > q and | (W0(z))| = q? The following result yields an
affirmative answer to these questions.
Theorem 1 Let p, q ∈ R, p > −1, 0 < q < π, and z ∈ C, z = 0. Then
(i) (W0(z)) < p if and only if either |z| < p exp(p) or
|z| ≥ |p| exp(p) and arccos
p exp(p)
|z|
+
|z|2 − p2 exp(2p)
exp(p)
< |Arg(z)| ;
(10)
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1550 J. ˇCermák et al.
(ii) (W0(z)) = p if and only if
|z| ≥ |p| exp(p) and arccos
p exp(p)
|z|
+
|z|2 − p2 exp(2p)
exp(p)
= |Arg(z)| ;
(11)
(iii) | (W0(z))| > q if and only if
|Arg(z)| > q and
q
sin(|Arg(z)| − q)
exp q cot(|Arg(z)| − q) < |z|; (12)
(iv) | (W0(z))| = q if and only if
|Arg(z)| > q and
q
sin(|Arg(z)| − q)
exp q cot(|Arg(z)| − q) = |z|. (13)
Proof (i) We write z = |z| exp(i Arg(z)) and put xk = (Wk(z)), yk = (Wk(z))
where Wk, k ∈ Z are particular branches of the Lambert function. Substitution into
(8) yields
exp(xk)(xk cos(yk) − yk sin(yk)) = |z| cos(Arg(z)), (14)
exp(xk)(xk sin(yk) + yk cos(yk)) = |z| sin(Arg(z)). (15)
If we solve (14)–(15) with respect to unknowns xk exp(xk) and yk exp(xk), then
xk exp(xk) = |z| cos(Arg(z) − yk), (16)
yk exp(xk) = |z| sin(Arg(z) − yk). (17)
To show that x0 = (W0)(z)) < p whenever |z| < p exp(p), we consider (16)
implying
x0 exp(x0) ≤ |z| < p exp(p).
Then the monotony property of the function g(p) = p exp(p) on (−1, ∞) actually
implies x0 < p.
Now we assume that |z| ≥ |p| exp(p). Squaring and adding (16) and (17) we get
|z|2
= ((xk)2
+ (yk)2
) exp(2xk),
i.e.,
|yk| =
|z|2 − (xk)2 exp(2xk)
exp(xk)
. (18)
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The Lambert function method in qualitative... 1551
For the principal branch, it holds x0 ≥ −y0 cot(y0), |y0| < π, i.e., x0 sin(y0) +
y0 cos(y0) ≥ 0 whenever y0 ≥ 0. Multiplying this by exp(x0) and using (15), one gets
|z| sin(Arg(z)) = exp(x0)(x0 sin(y0) + y0 cos(y0)) ≥ 0
which implies Arg(z) ≥ 0 for y0 ≥ 0. If y0 < 0, the same argumentation leads to
Arg(z) ≤ 0, hence Arg(z)y0 ≥ 0, i.e., | Arg(z) − y0| ≤ π. Then (16) with k = 0 is
equivalent to
arccos(x0 exp(x0)/|z|) = |Arg(z) − y0| . (19)
Moreover, sign analysis of (17) with respect to Arg(z)y0 ≥ 0 yields |Arg(z)| ≥ |y0|,
i.e.,
|Arg(z) − y0| = |Arg(z)| − |y0|. (20)
Then, using (18), (19) and (20), we are able to set up an implicit dependence between
x0 = (W0(z)) and z in the form f (x0, z) = 0 where f is defined via
f (p, z) = arccos
p exp(p)
|z|
− | Arg(z)| +
|z|2 − p2 exp(2p)
exp(p)
for all p > −1 and z ∈ C such that |p| exp(p) ≤ |z|. Let z be fixed. Then
d f
dp
(p, z) = −
(2p + 1) exp(3p) + |z|2 exp(p)
exp(2p) |z|2 − p2 exp(2p)
≤ −
(p + 1)2 exp(p)
|z|2 − p2 exp(2p)
≤ 0,
hence, f is decreasing in p if |p| exp(p) ≤ |z|. Therefore,
f (p, z) < f (x0, z) = 0
whenever
p > x0 = (W0(z)) and |p| exp(p) ≤ |z|.
(ii) The property follows directly from the proof of (i) using the fact that f (p, z) = 0
if and only if p = x0 due to monotony of f with respect to p.
(iii) Since W0 is symmetric in the sense W0(z) = W0(z) for all z ∈ C except those
lying on the branch cut along the negative real axis between −∞ and − exp(−1), it
suffices to assume the case y0 = (W0(z)) > q > 0. We have already observed that
Arg(z) ≥ y0. In addition, a stronger property holds, namely Arg(z) > y0. Indeed,
possible equality Arg(z) = y0 implies y0 = 0 (due to (17)) which contradicts the
assumption y0 > q > 0. Hence, it must be 0 < Arg(z) − y0 < π as well as
0 < Arg(z) − q < π. We divide (16) by (17) and put k = 0 to get
x0 = y0 cot(Arg(z) − y0). (21)
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1552 J. ˇCermák et al.
Taking the logarithm of (17) with k = 0, we also have
x0 = ln |z| sin(Arg(z) − y0) − ln(y0). (22)
Combining (21) and (22), we arrive at
ln |z| sin(Arg(z) − y0) − ln(y0) − y0 cot(Arg(z) − y0) = 0
representing again an implicit dependence, now between (W0(z)) and z. If we denote
h(q, z) = ln |z| sin(Arg(z) − q) − ln(q) − q cot(Arg(z) − q),
then we have
dh
dq
(q, z) =
−q sin(2(Arg(z) − q)) − sin2(Arg(z) − q) − q2
q sin2(Arg(z) − q)
.
While the denominator is positive, the numerator
N(q, z) = −q sin(2(Arg(z) − q)) − sin2
(Arg(z) − q) − q2
is negative for each 0 ≤ q ≤ Arg(z). Indeed, we have
dN
dq
(q, z) = 2q(cos2
(Arg(z) − q) − 1) ≤ 0
which implies that N(·, z) is non-increasing and, together with N(0, z) =
− sin2(Arg(z)) < 0, negative on [0, Arg(z)]. Consequently, h(·, z) is decreasing and
therefore h(q, z) > h(y0, z) = 0 whenever 0 < q < y0 < Arg(z). Taking into
account the above mentioned symmetry, we arrive (after some elementary algebra) at
(12).
(iv) The required property is again a consequence of monotony of the function h
from the previous part.
Remark 2 (a) The properties (ii) and (iv) of Theorem 1 provide a new tool for evaluations
of the principal branch of the Lambert function. Let z = 0 be a fixed complex
number. Then the left-hand side of (11) is decreasing for all p ∈ [a, W0(|z|)] (a = −1
if |z| ≥ exp(−1) and a = W0(−|z|) if |z| < exp(−1)) from π to the zero value. Hence,
(11) has a unique root p∗ lying in this interval, and this root equals just (W0(z)).
Similarly, the left-hand side of (13) is increasing for all q ∈ (0, Arg(z)) from the zero
value to infinity, i.e., (13) admits a unique positive root q∗ which is just (W0(z)).
To illustrate this evaluation technique, we compute W0(z) for z = 1
2 + i
√
3
2 . Then
|z| = 1, Arg(z) = π/3 and the standard Newton method returns (z) = p∗ ≈ 0.4843
in 5 iterations with the initial value p0 = 0.5 and the stopping criterion taken as
|pk+1 − pk| ≤ 10−16. The same method gives (z) = q∗ ≈ 0.3808 in 7 iterations
with the initial value q0 = 0.5 and the same precision as in the case of the real part.
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The Lambert function method in qualitative... 1553
In fact, the value p∗ + i q∗ matches the value produced by the MATLAB command
lambertw(1/2+sqrt(3)/2*1i) to all the 15 digits behind the decimal point.
Standardly,theNewtonorHalleymethodisapplieddirectlytotheequationw exp(w)−
z = 0 using the complex arithmetic. MATLAB employs the latter method with some
advanced guess of the starting point. For computing the values of the Lambert function
with arbitrary precision, we refer to the recent paper [14].
(b) Using a different approach, the property (i) of Theorem 1 was also discussed in
[21].
In the sequel, we clarify ordering of the real as well as imaginary parts of the
particular branches of the Lambert function. This ordering may be useful in a deeper
asymptotic analysis of (1), and, moreover, results into an alternate proof of Lemma 1.
Following the proof of Theorem 1, we introduce the functions
Gz(x, y) = x sin(y) + y cos(y) − |z| sin(Arg(z)) exp(−x) and
fz(x) = |z|2 exp(−2x) − x2.
In view of (15) and (18), the couples (xk, yk), where xk = (Wk(z)), yk = (Wk(z)),
have to meet the relations Gz(x, y) = 0 and y = ± fz(x), respectively.
The following assertion specifies ordering of imaginary parts of the branches of the
Lambert function.
Lemma 2 Let z ∈ C \ {0}. Then (Wk(z)) ≤ (Wk+1(z)) for all k ∈ Z. In fact, all
the inequalities are strict with the only exception: If z ∈ [− exp(−1), 0), then we have
(W−1(z)) = (W0(z)) = 0.
Proof For the sake of formal simplicity, we identify complex numbers w = x + i y
with couples (x, y) ∈ R2. First, let z ∈ C \ {0} be such that 0 ≤ Arg(z) ≤ π and
define sets Sz
j , j ∈ Z, as
Sz
j = {(x, y) ∈ R2 : Gz(x, y) = 0, (2 j − 1)π < y < (2 j + 1)π} for j = 1, 2, . . . ;
Sz
j = {(x, y) ∈ R2 : Gz(x, y) = 0, 0 ≤ y < π} for j = 0;
Sz
j = {(x, y) ∈ R2 : Gz(x, y) = 0, −2π < y ≤ 0} for j = −1;
Sz
j = {(x, y) ∈ R2 : Gz(x, y) = 0, 2 jπ < y < (2 j + 2)π} for j = −2, −3, . . .
(note that the equation Gz(x, y) = 0 has no solution for y = (2 j −1)π, j = 1, 2, . . . ,
and for y = 2 jπ, j = −1, −2, . . . ). We wish to show that Sz
j is a part of the range of
Wk just when j = k.
Let k ≥ 1 be arbitrary. Then, by the definition of Wk (see also Sect. 2),
(2k − 2)π < yk < (2k + 1)π. (23)
Let j be such that (xk, yk) ∈ Sz
j . We distinguish the following cases with respect to j.
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1554 J. ˇCermák et al.
If j > k, then yk > (2k + 1)π which contradicts (23). Let j = k. The ranges of
Wk and Wk+1, k = 0, 1, . . . , are separated by the curve
γk = {(u, v) ∈ R2
: u = −v cot(v), 2kπ < v < (2k + 1)π}
(see Sect. 2). The equation Gz(x, y) = 0, (2k − 1)π < y < (2k + 1)π, is equivalent
to
x = −y cot(y) + |z| sin(Arg(z)) exp(−x) (24)
provided y = 2kπ. To estimate the y-coordinate of a point (x, y) ∈ Sz
k , we put u = x
in γk.
First, let 2kπ < y < (2k + 1)π. Then any point (x, y) of Sz
k together with the
corresponding point (x, v) of γk have to fulfill the formula
y cot(y) − v cot(v) =
|z| sin(Arg(z))
sin(y)
exp(−x) (25)
due to x = −v cot(v) and (24). Since 0 ≤ Arg(z) ≤ π, the right-hand side of (25) is
non-negative, hence, we have y cot(y) ≥ v cot(v) implying y ≤ v. In other words, any
point (x, y) ∈ Sz
k is located below or on the curve γk separating Wk and Wk+1. Second,
let (2k −1)π < y ≤ 2kπ. Then the points (x, y) ∈ Sz
k lie below γk trivially (note that
the equation Gz(x, y) = 0 has a unique solution x for y = 2kπ, 0 < Arg(z) < π,
and has no solution for y = 2kπ, Arg(z) = 0 or Arg(z) = π). On the other hand, any
point of Sz
k lies above the upper bound (2k − 1)π of γk−1, hence, we have proven that
Sz
k is contained in the range of Wk.
Finally, if j < k, then we can similarly verify that any point of Sz
j is already located
below or on γk−1. Thus, to summarize the previous observations, Sz
j is contained in
the range of Wk (k = 1, 2, . . . ) just when j = k; otherwise, Sz
j and the range of Wk
are disjoint. Consequently, (2k − 1)π < yk < (2k + 1)π, hence, yk < yk+1 for all
k = 1, 2, . . ..
The same line of arguments can be used in the case k ≤ −1 to obtain 2kπ <
yk < (2k + 2)π, and, in the remaining case k = 0, to obtain 0 ≤ y0 < π (and
therefore, y0 < y1). In this respect, a real z is mapped by W0 to γ0 (hence, y0 > 0)
if z < − exp(−1), and is mapped by W0 to reals if z ≥ − exp(−1) (hence, y0 = 0).
Also W−1 takes real values just when − exp(−1) ≤ z < 0 (hence, y−1 = y0 = 0).
The procedure can be analogously applied to the case −π < Arg(z) < 0 (definition
of the sets Sz
j now differ by shifting the particular y-domains vertically down by
π).
Lemma 2 is useful also for ordering of the real parts of the branches of the Lambert
function. In particular, it enables to prove the classical assertion of Lemma 1 in a more
lucid way compared to the existing proof techniques.
Proof of Lemma 1 We recall that the couple (xk, yk), k ∈ Z, satisfies |y| = fz(x), see
the text preceding Lemma 2. Put ζ1 = W−1(−|z|), ζ2 = W0(−|z|), ζ3 = W0(|z|).
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The Lambert function method in qualitative... 1555
Then, we can easily observe that fz is defined on (−∞, ζ3] and (−∞, ζ1] ∪ [ζ2, ζ3] if
|z| ≥ exp(−1) and 0 < |z| < exp(−1), respectively. Moreover, fz has a (unique) root
ζ3 > 0 if |z| > exp(−1), a couple of roots ζ1 = ζ2 = −1 and ζ3 > 1 if |z| = exp(−1),
and a triple of roots ζ1 < −1, −1 < ζ2 < 0, ζ3 > 0 if 0 < |z| < exp(−1). Otherwise,
fz is positive at all other points of its domain.
Further, we have
fz (x) =
−|z|2 exp(−2x) − x
|z|2 exp(−2x) − x2
.
If |z| ≥
√
2
2 exp(−1/2), then fz is negative, hence fz is decreasing on (−∞, ζ3). If
√
2
2 exp(−1/2) > |z| > exp(−1), then fz has a local minimum at ζ4 = 1
2 W−1(−2|z|2)
and a local maximum at ζ5 = 1
2 W0(−2|z|2) (note that ζ4 < ζ5). Consequently, fz is
decreasing on (−∞, ζ4), increasing on (ζ4, ζ5) and again decreasing on (ζ5, ζ3).
If |z| = exp(−1), then fz is decreasing on (−∞, −1), increasing on (−1, ζ5)
and decreasing on (ζ5, ζ3). Finally, if exp(−1) > |z| > 0, then fz is decreasing on
(−∞, ζ1) increasing on (ζ2, ζ5) and decreasing on (ζ5, ζ3).
Thus, fz is decreasing on its domain up to “a small part” which is, however, lying
within the range of W0 (we again identify w = x + i y with (x, y) ∈ R2). Indeed, if√
2
2 exp(−1/2) > |z| > exp(−1), we have to show that the graph of fz between the
points [ζ4, fz(ζ4)] and [ζ5, fz(ζ5)] is contained in the range of W0. Since
ζ2
4 + f 2
z (ζ4) = −
1
2
W−1(−2|z|) < 1 and ζ2
5 + f 2
z (ζ5) = −
1
2
W0(−2|z|2
) <
1
2
for any
√
2
2 exp(−1/2) > |z| > exp(−1), both the endpoints of the graph belong
to the range of W0 (note that the open unit disk is a part of the W0 range). Also,
if ζ4 ≤ x ≤ ζ5, then x2 + f 2
z = |z| exp(−2x) is decreasing in x, hence, we have
|z| exp(−2x) < 1 for any ζ4 ≤ x ≤ ζ5 and any
√
2
2 exp(−1/2) > |z| > exp(−1)
meaning that the graph of the increasing part of fz is again lying in the range of W0.
Similarly, we can show that, for exp(−1) ≥ |z| > 0, the graph of fz between the
relevant points is contained in the range of W0 as well.
Collecting the above monotony properties together with Lemma 2, we can conclude
that x±(k+1) < x±k for any k = 1, 2, . . .. If k = 0, we have x1 < x0 but x−1 ≤ x0
because of W0(z) = W−1(z) for any z ∈ R, z ≤ − exp(−1).
Remark 3 We have actually proven monotony of the real parts of Wk(z) with respect
to k which is a slightly stronger result than that stated in Lemma 1.
4 Applications towards qualitative properties of (1) with the Caputo
derivative
In this section, we apply our previous observations on the principal branch of the
Lambert function to describe important qualitative properties of (1), including their
123
1556 J. ˇCermák et al.
Fig. 1 The set α,τ
0 as the stability region for (1) (the figure corresponds to α = 1.2)
dependence on the parameter λ. We remind that here we restrict ourselves to the case
α > 1 due to the reason mentioned in Remark 1.
We start with a basic stability criterion for (1). We introduce a (parametric) set α,τ
0
given as
α,τ
0 = z ∈ C : |z| <
|Arg(z)| − απ/2
τ
α
, |Arg(z)| >
απ
2
, (26)
see Fig. 1. This set already appeared in [6, Thm. 2] as the asymptotic stability region
for (1) with 0 < α < 1. As indicated in [7], the D-subdivision method (combined with
some tools of fractional calculus) used in [6] is extendable also to the case α > 1. In the
following assertion, we confirm validity of this stability result for α > 1. Contrary to
the existing techniques, we are able to prove this result (as a consequence of Theorem
1(i)) in an almost elementary way.
Theorem 2 Let α > 1, τ > 0 and λ ∈ C. Then (1) is asymptotically stable if and only
if λ ∈ α,τ
0 .
Proof By (9) and Lemma 1, we need to analyze (s0) = α
τ W0
τ
α λ1/α < 0. Using
(10) with z = τλ1/α/α and p = 0, this inequality can be converted into
τ
α
λ
1
α +
π
2
< Arg(λ
1
α ) ,
i.e.,
τ
α
|λ|
1
α < Arg(λ
1
α ) −
π
2
.
123
The Lambert function method in qualitative... 1557
Taking into account Arg(λ1/α) = 1
α |Arg(λ)|, this is equivalent to the condition
defining α,τ
0 in (26).
Remark 4 If λ ∈ ∂ α,τ
0 , then (1) is stable but not asymptotically stable. If λ /∈ cl α,τ
0 ,
then (1) is unstable. Obviously, α,τ
0 becomes empty for any α ≥ 2, hence, (1) cannot
be asymptotically stable for any complex λ whenever α ≥ 2.
Theorem 1 can be used in a more subtle way to bring a deeper insight into behavior
of (1). More precisely, relations (10)–(13) enable to reveal a relationship between the
position of the rightmost characteristic root s0 and the value of λ. Then, by (6), such
a relationship easily results into a precise asymptotic description of the solutions of
(1) in the unstable case ( (s0) > 0). Indeed, while in the asymptotically stable case
( (s0) < 0) the decay rate of solutions is algebraic and independent of the particular
value of s0, in the unstable case ( (s0) > 0), the growth rate is exponential and
governed by the real part of the rightmost characteristic root s0 (it is well known that
this is true also for integer values of α). In addition, the imaginary part of s0 is related
to the frequency characteristics describing an oscillatory behavior of (1).
Thus, for given u, v ≥ 0, we need to find a region of all complex λ such that the
rightmost characteristic root s0 is lying on or left to the line u + i ω, ω ∈ R, and on or
above (below) the line ω + i v (the line ω − i v), ω ∈ R. On this account, similarly as
in the proof of Theorem 2, we put z = τλ1/α/α, p = τu/α, q = τv/α and consider
u ≥ 0, απ/τ > v > 0. Then, Theorem 1 immediately implies that
(i) (s0) ≤ u if and only if either
|λ| < uα
exp(τu) (27)
or
|λ| ≥ uα
exp(τu) and α arccos
u exp(τu/α)
|λ|1/α
+
τ |λ|2/α − u2 exp(2τu/α)
exp(τu/α)
≤ |Arg(λ)| ; (28)
(ii) | (s0)| ≥ v if and only if
|Arg(λ)| > τv/α and
vα
sinα (|Arg(λ)| − τv)/α
× exp τv cot (|Arg(λ)| − τv)/α ≤ |λ|. (29)
Thus, if we introduce the set
α,τ
u as a set of all λ ∈ C such that either (27) or
(28) holds, and
α,τ
v as a set of all λ ∈ C such that (29) holds (we put
α,τ
v = ∅
for απ/τ ≤ v < π and
α,τ
v = C for v = 0), then we can rewrite our previous
observations into the following assertion:
Lemma 3 Let α > 1, τ > 0, u ≥ 0, π > v ≥ 0, λ ∈ C and let s0 be given by (9).
Then
123
1558 J. ˇCermák et al.
(i) (s0) ≤ u if and only if λ ∈
α,τ
u ;
(ii) | (s0)| ≥ v if and only if λ ∈
α,τ
v (
α,τ
v is non-empty whenever 0 ≤ v < απ/τ).
Remark 5 (a)Theset
α,τ
u containstheorigin(λ = 0)whichisexcludedbyTheorem1.
However, admitting λ = 0, we have the only characteristic root s0 = 0 of (2), hence,
Lemma 3 remains true.
(b) It is easy to check that α,τ
0 introduced in (26) coincides with int
α,τ
0 .
The part (i) of Lemma 3 immediately implies
Corollary 1 Let u ≥ 0 be fixed. Then
α,τ
u is the set of all λ ∈ C such that x(t) =
O(exp(ut)) as t → ∞ for any solution x of (1).
To obtain an actually effective (and non-improvable) asymptotic result for the solutions
of (1), we have to look at the problem inversely. More precisely, for a given
complex λ /∈ α,τ
0 , we need to find (non-negative) real values u0, v0 such that the
rightmost root s0 of (2) satisfies (s0) = u0, | (s0)| = v0.
The way how to do it easily follows from Theorems 1(ii) and 1(iv), respectively,
taking into account the above introduced substitutions z = τλ1/α/α, p = τu/α,
q = τv/α. Then the corresponding relations (11) and (13) become
|λ| ≥ uα
exp(τu) and α arccos
u exp(τu/α)
|λ|1/α
+
τ |λ|2/α − u2 exp(2τu/α)
exp(τu/α)
= |Arg(λ)| (30)
and
|Arg(λ)| > τv/α and
vα
sinα (|Arg(λ)| − τv)/α
exp τv cot (|Arg(λ)| − τv)/α
= |λ|, (31)
respectively (see also (28) and (29)). Thus, the unique solution of (30)2 defines the
value u0, while the unique positive solution of (31)2 defines the value v0 (provided
Arg(λ) > 0). If Arg(λ) = 0, then s0 is real, and we set v0 = 0.
Notice that (30)2, defining the relation between the modulus and argument of λ
that is explicit with respect to the argument, forms the boundary of
α,τ
u . Its lefthand
side, considered as a function of |λ|, is continuous, increasing and unbounded on
[uα exp(τu), ∞), and its graph in the complex plane creates a Jordan curve symmetric
with respect to the real axis, see Fig. 2.
Similarly, (31)2 provides the relation between the modulus and argument of λ that
is explicit with respect to the modulus. The equality (31)2 is the boundary of
α,τ
v , and
its left-hand side, as a function of |Arg(λ)|, is continuous on (τv/α, π] and unbounded
in a right neighborhood of the point τv/α (for |Arg(λ)| = π, it takes a value on the
negative real axis). This implies that
α,τ
v is unbounded for any 0 < v < απ/τ and
its boundary splits the complex plane into two parts, see Fig. 2.
Now we are in a position to formulate a complete asymptotic description for the
solutions of (1).
123
The Lambert function method in qualitative... 1559
Fig. 2 The figure depicts the boundaries of the sets
α,τ
u (blue) and
α,τ
v (orange) for several values of u
and v, respectively (the scenario corresponds to α = 1.2 and τ = 1). The particular blue curves represent
the set of all λ ∈ C such that the rightmost characteristic root s0 of (2) satisfies (s0) = u, and the particular
orange curves represent the set of all λ ∈ C such that the rightmost characteristic root s0 of (2) satisfies
| (s0)| = v. As an example, the blueish curvilinear rectangles then represent the set of all λ ∈ C such that
0.5 < (s0) < 0.75 and 1.25 < | (s0)| < 1.5
Theorem 3 Let α > 1, τ > 0 and λ ∈ C.
(i) If λ ∈ α,τ
0 , then, for any solution x of (1),
x(t) = O(t1−α
) as t → ∞.
Moreover the algebraic decay order 1 − α cannot be improved;
(ii) If λ /∈ α,τ
0 , then, for any solution x of (1),
x(t) = exp(u0t)(c exp(i v0t) + o(1)) as t → ∞
where c is a complex constant, u0 ≥ 0 is the unique solution of (30)2, v0 > 0 is
the unique solution of (31)2 if |Arg(λ)| > 0, and v0 = 0 if Arg(λ) = 0.
Proof (i) The property is a direct consequence of (6) as the set S is empty.
(ii) Let s0 = u0 + i v0 be the rightmost characteristic root, and S0 be the set of
the remaining characteristic roots s with a non-negative real part (we remind that
(s) < (s0) for any s ∈ S0). Then, if α is a non-integer, we can write (6) as
x(t) = exp(u0t) c exp(i v0t) +
s∈S0
cs exp((s − u0)t + O(t1−α
)
= exp(u0t) c exp(i v0t) + o(1) + O(o(1)) = exp(u0t) c exp(i v0t) + o(1)
123
1560 J. ˇCermák et al.
as t → ∞,
where c = cs0 is the complex constant from (6) corresponding to the rightmost characteristic
root s0. If α is an integer, then the dominating role of s0 in asymptotic behavior
of (1) is well known. In this case, the assertion of (ii) holds as well.
Remark 6 (a) The asymptotic formula from Theorem 3(ii) immediately implies
x(t) = O(exp(u0t)) as t → ∞ (32)
for any solution x of (1), and the constant u0 is non-improvable. Moreover, for large t,
the roots of the real and imaginary parts of x tend to the roots of cos(v0t) and sin(v0t),
respectively. In both the cases, the distance between the subsequent roots tends to
π/v0. These properties are illustrated by Example 1.
(b) The asymptotic behavior of (1) significantly depends on stability of (1). In
particular, the exponential terms in (6) are vanishing in the asymptotically stable
case (s0) < 0. However, the situation changes in the limit case α = 1 when, in
accordance with the first-order theory, the rightmost characteristic root s0 determines
an exponential decay rate of the solutions also in the asymptotically stable case. Since
the above argumentation can be extended to this problem as well, our results provide
a contribution also to the corresponding classical first-order theory.
Example 1 Let α = 1.2, τ = 1, and consider (1) along with the initial conditions
φ(t) = 1 (−1 ≤ t ≤ 0), φ0 = φ(0) = 1, and φ1 = limt→0+ x (t) = 0. We compare
the corresponding (numerical) solutions of (1) for two distinct values of λ, namely
λ1 = −2 + i and λ2 = −3 + i 0.1. As indicated by Fig. 2, both the values λ1, λ2 lie
in the instability region. In particular, the real parts u0 of the corresponding rightmost
roots are approximately 0.4721 and 0.4917, and their imaginary parts v0 are 1.2321
and 1.5844, respectively.
The real parts of the solutions of (1) with two above specified sets of entries, along
with the growth-rate functions exp(u0t), are depicted in Figs. 3 and 4. The graphs
suggest that the modulus of constant c introduced in Theorem 3(ii) is less than one for
λ = λ1, and greater than one for λ = λ2.
To illustrate behavior of the solutions x in better detail, Figs. 5 and 6 depict the ratio
(x(t))/ exp(u0t) for λ1 and λ2, respectively. The resulting functions are bounded,
but do not tend to zero which is a consequence of non-improvability of the constant
u0 in (32).
As mentioned in Remark 6(a), the distance between the subsequent roots of (x(t))
tends to π/v0. Figs. 7 and 8 illustrate this fact. We can see that while in the case of λ1
the convergence is rather fast and the distance seems to be somewhat stabilized around
the seventh root, in the case of λ2, the stabilization occurs around the hundredth root.
5 Concluding remarks
The aim of the paper was to develop the Lambert function theory, and then apply the
obtained results in qualitative investigations of (1). Using this approach, we were able
123
The Lambert function method in qualitative... 1561
Fig. 3 The real part of the solution x of (1) for α = 1.2, τ = 1 and λ1 = −2 + i, along with the
corresponding growth-rate functions ± exp(0.4721t)
Fig. 4 The real part of the solution x of (1) for α = 1.2, τ = 1 and λ2 = −3 + i 0.1, along with the
corresponding growth-rate functions ± exp(0.4917t)
to formulate a precise asymptotic description of the solutions of (1). Particularly, in
addition to an algebraic decay rate of the solutions in the stable case (described in
some earlier papers), we could observe an exponential growth of the solutions in the
123
1562 J. ˇCermák et al.
Fig. 5 The real part of the solution x of (1) for α = 1.2, τ = 1 and λ1 = −2 + i, divided by its growth-rate
function exp(0.4721t)
Fig. 6 The real part of the solution x of (1) for α = 1.2, τ = 1 and λ2 = −3 + i 0.1, divided by its
growth-rate function exp(0.4917t)
unstable case; the rate of this growth was determined as a (unique) real root of an
auxiliary transcendental equation.
123
The Lambert function method in qualitative... 1563
Fig. 7 The distance between the subsequent roots of (x(t)) for α = 1.2, τ = 1 and λ1 = −2+i is tending
to π/1.2321
Fig. 8 The distance between the subsequent roots of (x(t)) for α = 1.2, τ = 1 and λ2 = −3 + i 0.1 is
tending to π/1.5844
However, the impact of the presented results is not limited to the theory of FDDEs
only. Our approach offers an alternate way how to prove (and also strengthen) some
classical assertions of the Lambert function theory. Moreover, to the best of our knowledge,
the derived asymptotic formulae are new also in the first-order case. Here,
123
1564 J. ˇCermák et al.
contrary to the fractional case, our results can be applied also in the stable case where
a (non-improvable) rate of exponential decay of the solutions can be determined.
Since we have formulated our results for (1) with a complex coefficient λ, their
extension to the vector case is nearly straightforward provided the eigenvalues of
a (real) system matrix are simple. Regarding eigenvalues with higher multiplicities,
someadditionalargumentationseemstobenecessary.Basedonrelatedcasesdiscussed
in earlier papers, one can expect a slight modification of the solutions growth, but no
impact on the asymptotic frequency.
Our final remark concerns the case 0 < α < 1 not involved among the assumptions
of the assertions of Sect. 4. The procedure of computing the characteristic roots uses
the law of exponents which is, in general, not valid for complex numbers. Thus,
some superfluous roots of the characteristic equation may appear if 0 < α < 1 (as
illustrated via a counterexample in Remark 1). In this case, our stability and asymptotic
formulae remain basically true, but we cannot confirm their strictness. In particular,
we cannot claim that the above described rate of exponential growth of solutions is
non-improvable. Nevertheless, we conjecture that a more thorough analysis of the
corresponding branches of a complex power can overcome this problem, and thus
achieve the strict asymptotic results for all α > 0. Such an analysis provides another
possible topic for the next research.
Acknowledgements The research has been supported by the Grant GA20-11846S of the Czech Science
Foundation.
Funding Open access publishing supported by the National Technical Library in Prague.
Declarations
Conflict of interest The authors declare that they have no conflict of interest.
OpenAccess ThisarticleislicensedunderaCreativeCommonsAttribution4.0InternationalLicense,which
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123
Appendix E
Paper on lower-order two-term
FDDE [13] (AMC, 2017)
Fairly soon after entering the field of fractional delay equations, we began to deal
with linear equations involving fractional derivative, delayed term and also an undelayed
one. The addition of the undelayed term brings significant technical challenges,
even in scalar case, which we first addressed in [13] (co-authors: J. Čermák, Z. Došlá;
my author’s share 45 %).
We derived explicit necessary and sufficient conditions for asymptotic stability,
including asymptotic formulas for solutions (including algebraic decay rate towards
zero). To achieve this, we further developed our inverse Laplace transform technique
and introduced a broad family of functions containing exponentials, Mittag-Leffler
functions and generalized delay exponential of Mittag-Leffler type as special cases.
Although the stability region in space of equation’s coefficients appears simple,
it comprises a region of delay-independent stability and region where the stability
boundary depends on the delay. This work marked my first direct encounter with
the phenomenon of stability switching, where the stability property changes with
increasing delay. In this case, it meant just a one-time loss of stability, but in
subsequent research, much richer situations awaited us.
– 117 –
Applied Mathematics and Computation 298 (2017) 336–350
Contents lists available at ScienceDirect
Applied Mathematics and Computation
journal homepage: www.elsevier.com/locate/amc
Fractional differential equations with a constant delay:
Stability and asymptotics of solutions
Jan ˇCermáka,∗
, Zuzana Došláb
, Tomáš Kiselaa
a
Institute of Mathematics, Brno University of Technology, Technická 2, 616 69 Brno, Czech Republic
b
Department of Mathematics and Statistics, Faculty of Science, Masaryk University, Kotláˇrská 2, 611 37 Brno, Czech Republic
a r t i c l e i n f o
MSC:
34K37
34K20
34K25
Keywords:
Delay differential equation
Fractional-order derivative
Stability
Asymptotic behaviour
a b s t r a c t
The paper discusses stability and asymptotic properties of a fractional-order differential
equation involving both delayed as well as non-delayed terms. As the main results, explicit
necessary and sufficient conditions guaranteeing asymptotic stability of the zero solution
are presented, including asymptotic formulae for all solutions. The studied equation
represents a basic test equation for numerical analysis of delay differential equations of
fractional type. Therefore, the knowledge of optimal stability conditions is crucial, among
others, for numerical stability investigations of such equations. Theoretical conclusions are
supported by comments and comparisons distinguishing behaviour of a fractional-order
delay equation from its integer-order pattern.
© 2016 Elsevier Inc. All rights reserved.
1. Introduction
We investigate stability and asymptotic properties of the fractional delay differential equation
Dαy(t) = a y(t) + by(t − τ ), t > 0 (1)
with real coefficients a, b, a positive real lag τ and the fractional Caputo derivative operator Dα (0 < α < 1 is assumed to
be a real number).
Letting α → 1 from the left, Dαy(t) becomes y (t) and (1) is reduced to the classical delay differential equation
y (t) = a y(t) + by(t − τ ), t > 0 , (2)
studied frequently due to its theoretical as well as practical importance (see, e.g. [12]). This equation serves, among others,
as the basic test equation for stability analysis of various numerical discretizations of delay differential equations (see, e.g.
[1,9]). In this connection, stability conditions for (2) are traditionally required in the optimal form, i.e. as the necessary and
sufficient ones. There are known two types of such conditions for asymptotic stability of the zero solution of (2) that we
recall in the following two assertions (see [12]). As it is customary, by asymptotic stability of the zero solution of (2) we
understand the property that any solution y of (2) is eventually tending to the zero solution.
∗
Corresponding author.
E-mail addresses: cermak.j@fme.vutbr.cz (J. ˇCermák), dosla@math.muni.cz (Z. Došlá), kisela@fme.vutbr.cz (T. Kisela).
http://dx.doi.org/10.1016/j.amc.2016.11.016
0096-3003/© 2016 Elsevier Inc. All rights reserved.
J. ˇCermák et al. / Applied Mathematics and Computation 298 (2017) 336–350 337
Theorem 1. Let a, b and τ > 0 be real numbers. The zero solution of (2) is asymptotically stable if and only if the couple (a, b)
is an interior point of the area bounded by the line a + b = 0 from above and by the parametric curve
a = ϕ cot(τϕ), b = −
ϕ
sin(τϕ)
, ϕ ∈ 0,
π
τ
from below.
Theorem 2. Let a, b and τ > 0 be real numbers. The zero solution of (2) is asymptotically stable if and only if either
a ≤ b < −a and τ is arbitrary , (3)
or
|a| + b < 0 and τ <
arccos(−a/b)
(b2 − a2)1/2
. (4)
While Theorem 1 describes the stability boundary for (2) in the (a, b)-plane, the conditions of Theorem 2 seem to be
more explicit. In particular, (4)2 presents the value of the stability switch when (2) loses its stability property as the delay
τ is monotonically increasing.
It is also well-known that asymptotic stability of the zero solution of (2), described via the conditions of Theorems 1 and
2, is of exponential type, i.e. there exists δ > 0 such that y(t) = O(exp[−δt]) as t → ∞ for any solution y of (2). The value
δ depends on location of roots of the characteristic equation
s − a − bexp[−sτ] = 0 (5)
with respect to the imaginary axis (its estimates are discussed, e.g. in [8]).
The involvement of fractional-order derivatives into delay differential equations represents a new type combining advantages
of both delayed and non-integer derivative terms, especially hereditary properties, more degrees of freedom and
other advantages of fractional modelling. Since application areas of fractional delay differential equations are especially control
theory and robotics, the question of their stability (and asymptotics) is again of main interest. In general, stability and
asymptotic analysis of fractional delay differential equations is just at the beginning. As it is evident from the literature (see,
e.g. [6,16,17,21]), almost all the existing stability results on autonomous equations of this type are based on the root locus
of appropriate characteristic equations, and they do not provide universally acceptable effective criteria for testing stability
of a given fractional delay equation.
Therefore, the main goal of this paper is to extend the above stated properties of (2) to (1). Since (1) may serve as
a basic prototype of fractional delay differential equations, formulation of non-improvable stability conditions and related
asymptotic formulae is of a great importance in qualitative as well as numerical analysis of fractional delay differential
equations.
Following the classical case, we say that the zero solution of (1) is asymptotically stable if any solution y of (1) is eventually
tending to the zero solution. Similarly, the zero solution of (1) is called stable if any its solution is eventually bounded.
To describe asymptotics of solutions of (1), we shall use the following asymptotic symbols:
f (t) ∼ g(t) if lim
t→∞
|f (t)|
|g(t)|
= K > 0 ,
f (t) ∼sup g(t) if lim sup
t→∞
|f (t)|
|g(t)|
= K > 0 .
We note that (1) has been studied in the purely delayed case (when a = 0) in [5,15]. While the first paper presents
stability criterion based on a transcendent inequality involving the fractional Lambert function, the latter paper already
contains explicit conditions (which are, as we have already mentioned, very rare for fractional delay differential equations).
We recall here the explicit relevant results (reformulated to our notation) characterizing stability and asymptotics of (1) with
a = 0 (see [15, Theorem 5.1]).
Theorem 3. Let 0 < α < 1, b = 0 and τ > 0 be real numbers.
(i) The zero solution of
Dαy(t) = by(t − τ ), t > 0 (6)
is asymptotically stable if and only if
−
π − απ/2
τ
α
< b < 0.
In this case, y(t) ∼ t−α as t → ∞ for any solution y of (6).
(ii) The zero solution of (6) is stable, but not asymptotically stable, if and only if
b = −
π − απ/2
τ
α
.
338 J. ˇCermák et al. / Applied Mathematics and Computation 298 (2017) 336–350
A vector extension of Theorem 3 has been recently derived in [4]. As we have declared above, the goal of this paper
is to provide its another extension, namely to (1) involving both the delayed as well as non-delayed term (for some preliminary
stability results on (1) we refer to [2,11]). More precisely, we are going to derive direct fractional extensions of
Theorems 1 and 2, including asymptotic descriptions of solutions of (1).
The paper is organized as follows. Section 2 recalls some basic necessary notions and properties of fractional calculus,
especially those related to the Laplace transform. In Section 3, we discuss the characteristic equation associated with (1) and
describe some of its root properties. Section 4 presents main results on stability and asymptotics of (1). In particular, we formulate
here fractional analogues of Theorems 1 and 2 and present asymptotic formulae for solutions of (1). Some comments
and comparisons related to these results are mentioned as well. Section 5 is devoted to the proof of a key auxiliary result
describing asymptotics of (1) in terms of roots location of the associate characteristic equation. The final section summarizes
the results and outlines perspectives of a future research.
2. Preliminaries
Throughout this paper, we use the following standard definitions of the fractional integral of a real function f
D−ν f (t) =
t
0
(t − ξ)ν−1
(ν)
f (ξ)dξ, ν > 0, t > 0
and the Caputo fractional derivative of f
Dα f (t) = D−(1−α) d
dt
f (t) , 0 < α < 1, t > 0
where we put D0 f (t) = f (t) (for more details on basics of fractional calculus theory we refer, e.g. to [14,20,25]).
The main analytical technique used in stability investigations of linear fractional differential equations is based on the
well-known Laplace transform. For a given real function f, it is introduced via
L(f (t))(s) =
∞
0
f (t)exp[−st]dt ≡ F(s) , s ∈ D
where D ⊂ C contains all complex s such that the integral converges. The inverse Laplace transform of F can be expressed
via the contour integral
L−1
(F(s))(t) =
C
F(s)exp[st]ds
where the contour C ⊂ D is usually considered as a line R(s) = c such that all singularities of F lie to the left of this line.
However, in view of the fundamental theory of complex integration, this line can be changed into arbitrary contour C such
that its orientation with respect to singularities of F remains preserved (we recall that contour is an oriented piecewise
smooth curve). In our next analysis, we utilize the contour
γ (μ, ϑ) = γ1 + γ2 + γ3 (7)
such that its three oriented segments are given via
γ1 = {s ∈ C : s = −u exp[−iϑ], u ∈ (−∞, −μ)},
γ2 = {s ∈ C : s = μexp[iu], u ∈ [−ϑ, ϑ]},
γ3 = {s ∈ C : s = u exp[iϑ], u ∈ (μ, ∞)}
where μ > 0 and ϑ ∈ (−π, π]. This contour is depicted on Fig. 1.
A key computational property, namely the Laplace transform of fractional derivative of f, is given by the formula
L(Dα f (t))(s) = sαL(f (t))(s) − sα−1
f (0), 0 < α < 1 (8)
extending the classical first-order case (see, e.g. [20]). To avoid a possible misunderstanding, we emphasize that the principal
branch of the power functions occurring in (8) has to be considered. It is a consequence of the following property (for more
details see, e.g. [7, pp. 8–10]).
Proposition 1. Let q > −1. Then L(tq)(s) is a single-valued function given by
L(tq
)(s) = s−q−1
(q + 1)
where the principal branch of the complex power function is considered.
As the last step of this section, we introduce the family of functions
Ra,b,τ
α,β (t) = L−1 sα−β
sα − a − bexp[−sτ]
(t) (9)
J. ˇCermák et al. / Applied Mathematics and Computation 298 (2017) 336–350 339
Fig. 1. The contour γ (μ, ϑ) = γ1 + γ2 + γ3 in the complex plane.
where α, β, τ > 0 and a, b ∈ R. This family involves certain special functions appearing in fractional calculus such as
R0,0,τ
α,β (t) = tβ−1
/ (β),
Ra,0,τ
α,β (t) = tβ−1
Eα,β (atα ),
R0,b,τ
α,β (t) = Gb,τ,0
α,β (t),
Ra,b,0
α,β (t) = tβ−1
Eα,β ((a + b)tα ).
Here, Eα, β is the two-parameter Mittag–Leffler function (see, e.g. [20]) and Gb,τ,0
α,β is the generalized delay exponential function
of Mittag–Leffler type (see [4]). We can see that for α = β = 1 the functions coincide with the standard exponential
functions or delayed exponential functions known from the theory of delay differential equations (see, e.g. [3]).
The functions Ra,b,τ
α,β generalize the notion of fundamental solution for classical delay differential equations (see, e.g. [10]);
in particular, the representation formula
y(t) = φ(0)Ra,b,τ
α,1
(t) + b
0
−τ
Ra,b,τ
α,α (t − τ − u)h(t − τ − u)φ(u)du (10)
holds for all t > 0. Here, y is a solution of (1), h is the Heaviside step function and φ is the associated initial function (which
is supposed to be piecewise continuous on [−τ, 0]), i.e. it holds
y(t) = φ(t), −τ < t ≤ 0 . (11)
The verification of (10) is analogous to that employed in [4, Theorem 3] and therefore it is omitted. Based on (9) and (10),
Q(s) ≡ sα − a − bexp[−sτ] = 0 (12)
plays the role of a characteristic equation associated with (1). A more detailed explanation of this role and its connection to
stability properties of (1) is discussed in the next sections.
3. Distribution of roots of the characteristic equation
Eq. (12) admits infinitely many complex roots. In this section, we state some of their basic properties and analyse their
location in the complex plane.
Proposition 2. Let 0 < α < 1, a, b and τ > 0 be real numbers.
(i) If a + b ≥ 0 then (12) has a non-negative real root.
(ii) If s is a root of (12) then its complex conjugate ¯s is also a root of (12).
(iii) Let 0 < ω < π be arbitrary. Then (12) has no more than a finite number of roots s such that |arg(s)| ≤ ω.
(iv) If Q(s) = Q (s) = 0 for some complex s, then s is a positive real number.
340 J. ˇCermák et al. / Applied Mathematics and Computation 298 (2017) 336–350
Proof. If a + b ≥ 0 then Q(0) = −a − b ≤ 0. At the same time, Q(s) → ∞ as s → ∞, s being real. This implies the property
(i) (more precisely, (12) has the zero root if a + b = 0 and a positive real root if a + b > 0).
Now let
s = exp(iψ), (13)
= |s| and ψ = arg(s) ∈ (−π, π], be a root of (12). Then
α exp(iαψ) − a − bexp[− τ exp(iψ)] = 0 ,
i.e. ϱ and ψ satisfy
α cos(αψ) − a − bexp[− τ cos(ψ)] cos[ τ sin(ψ)] = 0 , (14)
α sin(αψ) + bexp[− τ cos(ψ)] sin[ τ sin(ψ)] = 0 . (15)
This immediately yields the property (ii). To prove (iii), we first show the existence of a real ϱ1 > 0 large enough such that
(12) has no roots s with |s| > ϱ1 and |arg(s)| ≤ ω. Squaring and adding (14) and (15) we get the relation
2α − 2a α cos(αψ) + a2
= b2
exp[−2 τ cos(ψ)] (16)
as the necessary condition for ϱ, ψ to satisfy (14) and (15). We distinguish two cases.
Let ω ≤ π/2. Then the property |arg(s)| ≤ ω implies |ψ| ≤ π/2, i.e.
b2
exp[−2 τ cos(ψ)] ≤ b2
.
On the other hand,
2α − 2a α cos(αψ) + a2
> ( α − |a|)2
.
This particularly implies that (16) cannot be satisfied for any ≥ 1, where
1 = (|a| + |b|)1/α ,
hence (12) has no roots with |s| ≥ 1
and |arg(s)| ≤ π/2.
Now let ω > π/2. Put κ = −τ cos(ω) > 0. Since
2α − 2a α cos(αψ) + a2
< ( α + |a|)2
,
it is enough to find 1 > 0 such that
α + |a| < |b|exp[κ ]
for all ≥ 1 . The existence of such a value is obvious and, using appropriate elementary calculations, we can give various
specifications of 1
. To summarize it, for any given 0 < ω < π, we can find ρ1 > 0 such that (12) has no roots s with |s| >
ϱ1 and |arg(s)| ≤ ω.
Further, we can easily observe the existence of a real ϱ2 > 0 sufficiently small such that (12) has no nonzero roots lying
inside the circle |s| < ϱ2. Indeed, the Taylor expansion enables to write (12) as
sα = a + b(1 − sτ + s2
τ2
/2! − · · · ). (17)
If a + b = 0 then (17) has no roots s with |s| < ϱ2, ϱ2 > 0 being sufficiently small. If a + b = 0, then s = 0 is a root of (17) and,
assuming s = 0, (17) yields
sα−1
= −b(τ − sτ2
/2! + · · · ),
hence (17) has no nonzero roots s with |s| < ϱ2, ϱ2 > 0 being sufficiently small.
Finally, let be the compact set of complex s such that ϱ1 ≤ |s| ≤ ϱ2 and |arg(s)| ≤ ω. Then Q is analytic on , hence
it cannot have infinitely many roots in . This completes the proof of (iii).
To prove the property (iv), we assume that the relations
sα − a − bexp[−sτ] = 0 and αsα−1
+ τbexp[−sτ] = 0 (18)
hold for a suitable nonzero complex s (obviously, s = 0 cannot satisfy (18)). Elimination of the exponential term yields
a = sα−1
s +
α
τ
. (19)
If we substitute (13) into (19) and compare imaginary parts, we obtain
τ sin(αψ) = α sin[(1 − α)ψ] . (20)
Similarly, (18) implies
b = −
α
τ
sα−1
exp[sτ] (21)
J. ˇCermák et al. / Applied Mathematics and Computation 298 (2017) 336–350 341
and an analogous argumentation as used above leads to
τ sin(ψ) + (α − 1)ψ = kπ (22)
for a suitable integer k.
Now assume that ψ = 0. Then we can express τϱ from (20) and (22) to compare their corresponding right-hand sides.
This yields
g1(ψ) ≡
α sin(ψ)
sin(αψ)
=
(1 − α)ψ + kπ
sin[(1 − α)ψ]
≡ g2(ψ). (23)
We show that
g1(ψ) < g2(ψ) for all −π < ψ ≤ π, ψ = 0 and all integers k. (24)
Obviously, it is enough to check (24) for all 0 < ψ ≤ π and k = 0. In such a case, g1(ψ) = g2(ψ) = 1 as ψ → 0 and g1(ψ) <
0, g2
(ψ) > 0 for all 0 < ψ < π (verification of these sing derivative properties requires some routine and straightforward
calculations which are omitted). This confirms (24), hence (23) cannot occur for any −π < ψ ≤ π, ψ = 0 and any integer
k. In other words, (18) cannot be satisfied if s is a complex number with a nonzero argument ψ.
If ψ = 0 then s = and
a = α−1
+
α
τ
, b = −
α
τ
α−1
exp[ τ] (25)
by use of (19) and (21). For all ϱ > 0, (25) defines the set of all couples (a, b) such that there exists a positive real s satisfying
(18).
Remark 1.
(a) We can easily check that the condition Q(s) = Q (s) = Q (s) = 0 does not hold for any complex s. In other words, if
s is a multiple root of Q, then it is a (positive real) double root.
(b) The property Proposition 2 (iii) generalizes the well-known fact, namely that (5) has only finitely many roots with
positive real parts (indeed, if we put ω = π/2, we get such a conclusion also for (12)).
In classical analysis of (2), asymptotic stability of the zero solution occurs if and only if all roots of (5) have negative
real parts (effective conditions ensuring this property are involved in Theorems 1 and 2). In the next section, we deduce an
analogous requirement on distribution of roots of (12). Therefore, the next aim of this section is to derive effective conditions
on parameters of (12) ensuring that all its roots have negative real parts.
In the case of standard (integer-order) delay differential equations and their associated characteristic equations, this matter
is solved via the D-partition method (see, e.g. [12]). This method is applicable also in the fractional-order case and we
perform its basic steps (routine calculations will be omitted).
Let BLτ
α(a, b) be the boundary locus for (12), i.e. the set of all real couples (a, b) such that (12) admits a purely imaginary
root s = iϕ = ϕ exp[iπ/2] (note that it is enough to consider only the case ϕ ≥ 0 due to the property Proposition 2 (ii)). By
(14) and (15), such couples (a, b) satisfy
ϕα cos(απ/2) − a − bcos(ϕτ ) = 0 , (26)
ϕα sin(απ/2) + bsin(ϕτ ) = 0 . (27)
Now we distinguish two cases. First let ϕτ = mπ for a suitable integer m. Then (27) implies ϕ = 0, i.e. m = 0, and the
relevant part of BLτ
α(a, b) corresponding to this case consists of the line a + b = 0.
Now let ϕτ = mπ for all integers m. Then (26) and (27) yield the solution
a =
ϕα sin(ϕτ + απ/2)
sin(ϕτ )
, b = −
ϕα sin(απ/2)
sin(ϕτ )
.
Hence, BLτ
α(a, b) corresponding to this case is formed by the system of curves
am(ϕ) =
ϕα sin(ϕτ + απ/2)
sin(ϕτ )
, bm(ϕ) = −
ϕα sin(απ/2)
sin(ϕτ )
, (28)
mπ/τ < ϕ < (m + 1)π/τ, m = 0, 1, . . .. The set BLτ
α(a, b) is depicted in the (a, b)-plane on Fig. 2. We note that the construction
of this set for (12) involving multi-valued function sα has been performed in [11]. As we have emphasized in the part
preceding Proposition 1, it is enough to consider appropriate single-valued function in this problem.
Our next analysis of the area of couples (a, b) such that (12) has all roots with negative real parts originates from
continuous dependance of roots of (12) on the coefficients a, b. This property particularly implies that the number of roots
of (12) with positive real parts can be changed only when its coefficients a, b are crossing an element of BLτ
α(a, b). In other
words, the number of roots of (12) having a positive real part remains unchanged in all open sets whose boundaries are
formed by the line a + b = 0 or by some curves (28) (or their parts). Then it is enough to choose representatives of these
open sets to specify the number of roots of (12) with positive real parts within these sets.
342 J. ˇCermák et al. / Applied Mathematics and Computation 298 (2017) 336–350
Fig. 2. The set BLτ
α (a, b) for α = 0.4, τ = 1.
Since we are looking for the couples (a, b) such that (12) has all roots with negative real parts, Proposition 2 (i) implies
that it is enough to restrict on the half-plane a + b < 0. It is easy to check that the curves (28) with even m intersect the
line a + b = 0 when ϕ = (m + 1 − α)π/τ, hence it remains to investigate their parts
(am(ϕ), bm(ϕ)), (m + 1 − α)π/τ < ϕ < (m + 1)π/τ, m = 0, 2, 4, . . .
lying to the left of this line. A more detailed computational analysis shows that these curves cross the b-axis at the points
b∗
m = −[(2m + 2 − α)π/(2τ )]α, tend to the asymptotes b = a − [(m + 1)π/τ]α cos(απ/2) as ϕ → (m + 1)π/τ from the left
and, moreover, these curves do not intersect each other (see also Fig. 2). Then, choosing appropriate points on the b-axis
as suitable representatives, previous considerations along with Theorem 3 yield that (12) has all roots with negative real
parts if and only if the couple (a, b) is an interior point of the area bounded by the line a + b = 0 from above and by the
parametric curve
a =
ϕα sin(τϕ + απ/2)
sin(τϕ)
, b = −
ϕα sin(απ/2)
sin(τϕ)
, ϕ ∈
(1 − α)π
τ
,
π
τ
(29)
from below.
Note also that an alternative way how to deduce such an area from a given boundary locus is based on differentiation of
(12) with respect to some of its parameters. Following this way, it can be shown that d(R(s))/da > 0 at s = iϕ. This implies
that all roots of (12) cross the imaginary axis at s = iϕ from the left to the right as a increases, and thus we arrive at the
same conclusion as above.
We can summarize the previous considerations in the following
Proposition 3. Let 0 < α < 1, a, b and τ > 0 be real numbers. Then all roots of (12) have negative real parts if and only if the
couple (a, b) is an interior point of the area bounded by the line a + b = 0 from above and by the parametric curve (29) from
below.
Now we rewrite the parametric form (29) into the explicit one. Doing this, (29) implies that
a
−b
=
sin(ϕτ + απ/2)
sin(απ/2)
.
Taking into account the restriction ϕ ∈ ((1 − α)π/τ, π/τ ), we can express ϕ and eliminate it via its substitution into (29)2.
This yields
b = −
(1 − α)π/2 + arccos[(−a/b)sin(απ/2)]
α
sin(απ/2)
τα sin (1 − α)π/2 + arccos[(−a/b)sin(απ/2)]
.
Obviously, some additional calculations enable to determine the delay τ from this relation explicitly. Consequently, this
argumentation implies that (29) can be written as τ = τ∗ where
τ∗
=
(1 − α)π/2 + arccos[(−a/b)sin(απ/2)]
[a cos(απ/2) + (b2 − a2 sin
2
(απ/2))1/2]1/α
(30)
and |a| + b < 0. Using this notation we have
J. ˇCermák et al. / Applied Mathematics and Computation 298 (2017) 336–350 343
Proposition 4. Let 0 < α < 1, a, b and τ > 0 be real numbers. Then all roots of (12) have negative real parts if and only if it
holds either
a ≤ b < −a and τ is arbitrary , (31)
or
|a| + b < 0 and τ < τ∗
. (32)
Remark 2. A similar issue has been discussed in [2] where (12) is analysed under the restrictive assumption a < 0 (in our
notation). The computational technique used in [2] is different from ours and leads to a formally slightly more complicated
evaluation of τ∗.
4. Stability and asymptotics of solutions
This section presents our main results on (1). First, we formulate the auxiliary assertion providing a key tool in stability
and asymptotic analysis of (1).
Lemma 1. Let 0 < α < 1, a, b and τ > 0 be real numbers and let y be a solution of (1).
(i) Let all roots of (12) have negative real parts. Then
y(t) ∼ t−α or y(t) = O(t−α−1
) as t → ∞ . (33)
(ii) Let all roots of (12) have non-positive real parts and let there exist a root with the zero real part. If (12) has no purely
imaginary roots, then
y(t) ∼ 1 or y(t) = O(tα−1
) as t → ∞ . (34)
If (12) has purely imaginary roots, then
y(t) ∼sup 1 or y(t) = O(1) as t → ∞ . (35)
(iii) Let there exist a root of (12) with a positive real part. Then y(t) = O(t exp[Mt]) as t → ∞, where M = maxsi
(R(si)), si
being roots of (12). Moreover, there exists a solution y of (1) such that
y(t) ∼sup t exp[Mt] or y(t) ∼sup exp[Mt] as t → ∞ . (36)
Remark 3. The asymptotic property (33)1 or (33)2 occurs if φ(0) = 0 or φ(0) = 0, respectively, where φ is the initial function
implying a particular solution of (1) via (11). The same comment holds also for (34) and (35) (a more detailed analysis
is involved in Section 5). In particular, under the assumption of the part (ii), there always exist solutions of (1) not tending
to the zero solution. Regarding (36), the type of asymptotic behaviour depends on multiplicity of the root of (12) with the
maximal positive real part (see also Proposition 2 (iv) and Remark 1).
Now we are in a position to formulate our main stability and asymptotic criteria on (1). If we put through Propositions 3,
4 and Lemma 1, we arrive at the following two results.
Theorem 4. Let 0 < α < 1, a, b and τ > 0 be real numbers.
(i) The zero solution of (1) is asymptotically stable if and only if the couple (a, b) is an interior point of the area bounded by
the line a + b = 0 from above and by the parametric curve (29) from below. In this case, (33) holds for any solution y of
(1).
(ii) The zero solution of (1) is stable, but not asymptotically stable, if and only if either
a + b = 0, a ≤
[π(1 − α)]α
2τα cos(απ/2)
, (37)
or a, b satisfy (29) for an admissible value ϕ. In this case, (34) or (35) holds for any solution y of (1).
Theorem 5. Let 0 < α < 1, a, b and τ > 0 be real numbers.
(i) The zero solution of (1) is asymptotically stable if and only if (31) or (32) is satisfied. In this case, (33) holds for any
solution y of (1).
(ii) The zero solution of (1) is stable, but not asymptotically stable, if and only if either (37), or
|a| + b < 0, τ = τ∗
, τ∗
being given by (30)
is satisfied. In this case, (34) or (35) holds for any solution y of (1).
344 J. ˇCermák et al. / Applied Mathematics and Computation 298 (2017) 336–350
Fig. 3. The stability region for (1) with α = 0.4 and τ = 1.
The stability area for (1), described via the conditions of Theorems 4 and 5, is depicted in the (a, b)-plane on Fig. 3.
Remark 4. The assertions of Theorems 4 and 5 provide a direct extension of Theorems 1 and 2, respectively. Indeed, letting
α → 1 from the left, one can observe a full agreement between appropriate formulae for the fractional and classical case.
In particular, the value (30) of the fractional stability switch, when asymptotic stability of the zero solution of (1) is turning
into instability as the delay τ is monotonically increasing, becomes the appropriate value from (4). Notice also, that if we
put a = 0 in Theorem 5, we obtain just the assertion of Theorem 3 (where the condition φ(0) = 0 is considered).
Now we make some comments on two qualitative dissimilarities between the fractional and classical delayed case. First,
the decay rate of solutions in the asymptotically stable case is exponential when α = 1, but only algebraic when 0 < α < 1.
Second, the situation on the stability boundary differs at the cusp point
Pα =
[π(1 − α)]α
2τα cos(απ/2)
; −
[π(1 − α)]α
2τα cos(απ/2)
where the line a + b = 0 intersects the transcendental curve (29). The cusp point Pα corresponds to the zero (simple) root
of (12) and Theorem 4 (ii) (as well as Theorem 5 (ii)) immediately yields
Corollary 1. Let 0 < α < 1 and τ > 0 be real numbers. The zero solution of
Dαy(t) =
[π(1 − α)]α
2τα cos(απ/2)
[y(t) − y(t − τ )], t > 0 (38)
is stable, but not asymptotically stable.
Letting α → 1 from the left, (38) turns into
y (t) =
1
τ
[y(t) − y(t − τ )], t > 0 (39)
and the cusp point Pα becomes P = [1/τ; −1/τ]. A direct procedure shows that P corresponds to the double zero root of
(5) with a = −b = 1/τ, hence the zero solution of (39) cannot be stable. Indeed, we can easily check that (39) admits an
unbounded solution, namely y(t) = t. Consequently, the stability property of (38) is not transferable to its limit case (39).
Thus, this example confirms a positive impact of derivatives of real orders between 0 and 1 on stability properties of studied
delay equation.
5. Proof of Lemma 1
First, we present several technical assertions where we use the notation
σ = γ (ε/t, π/2 + δ), t > 0 , (40)
σ = γ (εα, απ/2 + αδ) (41)
with real constants ε, δ > 0, 0 < α < 1 and the contour γ given by (7).
J. ˇCermák et al. / Applied Mathematics and Computation 298 (2017) 336–350 345
Proposition 5. For given 0 < α < 1 and τ > 0, we denote
ωx,n(t) =
1
2παi σ
ux
exp[u1/α(1 + nτ/t)]
(u − atα )exp[u1/ατ/t] − btα
du , x > −1, n = 2, 3, . . . .
(i) If there exists η0 > 0 such that
|(u − atα )exp[u1/ατ/t] − btα| ≥ η0|exp[u1/ατ/t]| (42)
for all u ∈ σ , then
ωx,n(t) = O(1) as t → ∞ . (43)
(ii) If there exists η1 > 0 such that
|(u − atα )exp[u1/ατ/t] − btα| ≥ η1tα|exp[u1/ατ/t]| (44)
for all u ∈ σ , then
ωx,n(t) = O(t−α ) as t → ∞ , (45)
ωx,n+m(t) = ωx,n(t) + O(t−1−α ) as t → ∞ (46)
where m ∈ Z+ is arbitrary.
Proof. First, we prove (43) and (45) simultaneously. Let κ ∈ {−α, 0}. Utilizing (42) and (44), we obtain
|ωx,n(t)| ≤
tκ
2παη0 σ
|ux
||exp [u1/α(1 + (n − 1)τ/t)]||du|
≤
tκ
2παη0
απ/2+αδ
−απ/2−αδ
εα(x+1)
exp[ε(1 + (n − 1)τ/t)cos(ϕ/α)]dϕ
+ 2
∞
εα
rx
exp[r1/α(1 + (n − 1)τ/t)cos(π/2 + δ)]dr
≤
tκ
2παη0
α(π + 2δ) +
α (α(x + 1))
(cos(π/2 − δ))α(x+1)
= O(tκ ) as t → ∞
where we have employed the integral definition of the Gamma function (computational details are omitted). This proves
(43) and (45).
The formula (46) can be derived via a direct calculation:
ωx,n+m(t) =
1
2παi σ
ux
exp [u1/α(1 + nτ/t)]
(u − atα )exp[u1/ατ/t] − btα
∞
j=0
(mτ )j
j!t j
uj/αdu
= ωx,n(t) +
1
2παi
∞
j=1
(mτ )j
j!t j
σ
ux+ j/α exp [u1/α(1 + nτ/t)]
(u − atα )exp[u1/ατ/t] − btα
du
= ωx,n(t) +
∞
j=1
(mτ )j
j!t j
ωj/α+x,n(t) = ωx,n(t) + O(t−α−1
) as t → ∞ .
Proposition 6. For given 0 < α < 1 and τ > 0, we denote
θx,n(t) =
1
2παi σ
ux
exp[u1/α(1 + nτ/t)]du , x > −1, n = 2, 3, . . . .
Then
θx,n(t) =
tαx+α(t + nτ )−αx−α
(1 − αx − α)
.
Proof. We set v = u(1 + nτ/t)α to get
θx,n(t) =
1
2παi
t
t + nτ
αx+α
γ ((1+nτ/t)αεα,απ/2+αδ)
vx
exp[v1/α]dv.
Applying the formula
1
(z)
=
1
2παi γ (r,ψ)
exp[ξ1/α]ξ1/α−z/α−1
dξ, r > 0, πα/2 < ψ < απ
(see, e.g. [20]), we obtain the result.
346 J. ˇCermák et al. / Applied Mathematics and Computation 298 (2017) 336–350
Now we can formulate the key auxilliary assertion of this section dealing with asymptotic properties of the class of Ra,b,τ
α,β
functions.
Lemma 2. Let 0 < α < 1, 0 < β ≤ 1, a, b and τ > 0 be real numbers and let si be roots of (12).
(i) If R(si) < 0 for all si then
Ra,b,τ
α,β (t) ∼ tβ−α−1
for α = β and Ra,b,τ
α,α (t) = O(t−α−1
) as t → ∞ .
(ii) If there exists the zero root of (12) and R(si) < 0 otherwise, then
Ra,b,τ
α,1
(t) ∼ 1 and Ra,b,τ
α,β (t) = O(tβ−1
) for β < 1 as t → ∞ .
(iii) If R(si) ≤ 0 for all si and some of the roots are purely imaginary, then
Ra,b,τ
α,β (t) ∼sup 1 as t → ∞ .
(iv) If R(si) > 0 for some si then
Ra,b,τ
α,β (t) ∼sup (Bt + C)exp[Mt] as t → ∞
where M = maxsi
(R(si)) and reals B, C ≥ 0 are such that B + C > 0.
Proof. It is enough to consider the case b = 0 (for the case b = 0 we refer to [18]). Proposition 2 (iii) implies that there
exists δ > 0 such that all nonzero roots si of (12) satisfy |arg(si)| = π/2 + δ and, moreover, there are only finitely many of
them satisfying |arg(si)| < π/2 + δ. Consequently, for every t > 1, we can choose R > ε > 0 such that all si lie to the left
of γ (R, π/2 + δ) and those satisfying |arg(si)| < π/2 + δ are located to the right of σ given by (40). Thus, we can split the
inverse Laplace transform formula into
Ra,b,τ
α,β (t) =
1
2πi γ (R, π
2 +δ)
sα−β exp[ts]
sα − a − bexp[−τs]
ds = I1(t) + I2(t),
where I1 and I2 denote integrals over γ (R, π/2 + δ) − σ and σ, respectively.
I. First, we analyse I1. Clearly, the contour γ (R, π
2 + δ) − σ is a simple positively oriented closed curve. Proposition 2 (iii)
implies that the integrand has only finitely many poles si (i = 1, . . . , N) lying in the interior of γ (R, π
2 + δ) − σ. The residue
theorem yields
I1(t) =
1
2πi γ (R, π
2 +δ)−σ
sα−β exp[st]
sα − a − bexp[−τs]
ds =
N
i=1
Ress=si
sα−β exp[st]
sα − a − bexp[−τs]
.
Proposition 2 (iv) shows that all poles si with nonzero imaginary parts are simple, while real poles (more precisely, positive
real poles) admit double multiplicity. Thus, for every pole si, we can utilize the Laurent expansions
sα−β
sα − a − bexp[−τs]
= ai
−2(s − si)−2
+ ai
−1(s − si)−1
+ ai
0 + ai
1(s − si) + ai
2(s − si)2
+ · · · ,
exp[st] = exp[sit] 1 + t(s − si) +
t2
2!
(s − si)2
+
t3
3!
(s − si)3
+ · · ·
where aj ( j = −2, −1, 0 . . .) are complex constants independent of t. Hence, a multiplication of both expansions enables us
to write the sum of residues (i.e. the coefficients at (s − si)−1) as
I1(t) =
N
i=1
(ai
−1 + ai
−2t)exp[sit] ,
where ai
−1
, ai
−2
(i = 1, . . . , N) are complex constants such that for any i at least one of the terms is nonzero.
II. Now, we analyse the term I2. Employing the change of variables s = u1/α/t, which transforms the contour σ into σ
(given by (41)), we get
I2(t) =
1
2πi σ
sα−β exp[st]
sα − a − bexp [−τs]
ds = tβ−1
ω(1−β)/α,1(t) (47)
where ω(1−β)/α,1 is introduced in Proposition 5.
II-1. Let (12) have the zero root, i.e. a + b = 0. Although σ is chosen so that it does not contain any roots of (12), it
approaches the zero root as t → ∞. The closest points s of σ with respect to the root s = 0 satisfy s = ε/t exp[iϕ], i.e. we
can write
|sα − a − bexp[−τs]| ≥ |εαt−α exp[iαϕ] − a − bexp[−ετ/t]| ≥ η0t−α
J. ˇCermák et al. / Applied Mathematics and Computation 298 (2017) 336–350 347
for a suitable η0 > 0 and sufficiently large t. Applying the change of variable s = u1/α/t, we can directly employ (43) to
obtain the asymptotic estimate
I2(t) = O(tβ−1
) as t → ∞ .
II-2. Let (12) have only nonzero roots, i.e. a + b = 0. Since the contour σ does not contain any root si of (12), there exists
η1 > 0 such that
|sα − a − bexp[−τs]| ≥ η1
for all s ∈ σ. Similarly to II-1, as a direct consequence of (45) we get
I2(t) = O(tβ−α−1
) as t → ∞ .
In the sequel, we precise this estimate.
II-2-a. Let a = b. Substituting the above suggested change of variable, we get (along with some simple calculations) that
(44) is satisfied. Now, we use the identity
1
ξ − z
= −
1
z
−
ξ
z2
+
ξ2
z2(ξ − z)
(48)
(with z = btα and ξ = (u − atα )exp[u1/ατ/t]) and (45), (46) to get
I2(t) = −
tβ−α−1
b
θ(1−β)/α,1(t) −
tβ−2α−1
b2
θ(1−β)/α+1,2(t) +
atβ−α−1
b2
θ(1−β)/α,2(t)
+
tβ−2α−1
b2
σ
u(1−β)/α(u − atα )2
exp[u1/α(1 + 3τ/t)]
(u − atα )exp[u1/ατ/t] − btα
du .
Further decomposition of the last term yields
I2(t) = −
θ(1−β)/α,1(t)
bt−β+α+1
−
θ(1−β)/α+1,2(t)
b2t−β+2α+1
+
aθ(1−β)/α,2(t)
b2t−β+α+1
+
ω(1−β)/α+2,3(t)
b2t−β+2α+1
−
2aω(1−β)/α+1,3(t)
b2t−β+α+1
+
a2
ω(1−β)/α,3(t)
b2t−β+1
= −
θ(1−β)/α,1(t)
bt−β+α+1
−
θ(1−β)/α+1,2(t)
b2t−β+2α+1
+
aθ(1−β)/α,2(t)
b2t−β+α+1
+
ω(1−β)/α+2,3(t)
b2t−β+2α+1
−
2aω(1−β)/α+1,3(t)
b2t−β+α+1
+
a2
b2
I2(t) + O(tβ−α−2
)
as t → ∞. Rearranging the expression, we arrive at
I2(t) =
b2
b2 − a2
−
θ(1−β)/α,1(t)
bt−β+α+1
−
θ(1−β)/α+1,2(t)
b2t−β+2α+1
+
aθ(1−β)/α,2(t)
b2t−β+α+1
+
ω(1−β)/α+2,3(t)
b2t−β+2α+1
−
2aω(1−β)/α+1,3(t)
b2t−β+α+1
+ O(tβ−α−2
)
as t → ∞. An analogous argumentation yields
ω(1−β)/α+1,3(t) =
b2
b2 − a2
−
θ(1−β)/α+1,3(t)
btα
−
θ(1−β)/α+2,4(t)
b2t2α
+
aθ(1−β)/α+1,4(t)
b2tα
+
ω(1−β)/α+3,5(t)
b2t2α
−
2aω(1−β)/α+2,5(t)
b2tα
+ O(t−α−1
)
as t → ∞. Combining these expressions with Propositions 5 (ii) and 6, we get
I2(t) =
b2
b2 − a2
−
(t + τ )β−α−1
b (β − α)
−
(t + 2τ )β−2α−1
b2 (β − 2α)
+
a(t + 2τ )β−α−1
b2 (β − α)
+ O(tβ−3α−1
) +
2a(t + 3τ )β−2α−1
b(b2 − a2) (β − 2α)
+
2a(t + 4τ )β−3α−1
b2(b2 − a2) (β − 3α)
−
2a2
(t + 4τ )β−2α−1
b2(b2 − a2) (β − 2α)
+ O(tβ−4α−1
) + O(tβ−3α−1
) + O(tβ−2α−2
) + O(tβ−α−2
)
=
b(t + τ )β−α−1
− a(t + 2τ )β−α−1
−(b2 − a2) (β − α)
+
(b2
− a2
)(t + 2τ )β−2α−1
− 2ab(t + 3τ )β−2α−1
+ 2a2
(t + 4τ )β−2α−1
−(b2 − a2)2 (β − 2α)
+ O(tβ−α−1−min(2α,1)
)
348 J. ˇCermák et al. / Applied Mathematics and Computation 298 (2017) 336–350
as t → ∞. Taking into account the property 1/ (0) = 0, we can derive
lim
t→∞
|I2(t)|
tβ−α−1
=
−(a + b)−1
(β − α)
and lim
t→∞
|I2(t)|
tβ−2α−1
=
−(a + b)−2
(β − 2α)
(49)
for α = β and α = β, respectively. Thus, the asymptotic behaviour of I2 is described for a = b.
II-2-b. Now, let a = b = 0 and α = β. We show that (49)1 is transferable to this case through a = b + ζ as ζ → 0. Based
on (47) and (49)1, we denote
I
ζ
2
(t) =
tβ−1
2παi σ
u(1−β)/α exp[u1/α(1 + τ/t)]
(u − (b + ζ )tα )exp[u1/ατ/t] − btα
du ,
L(ζ ) =
−1
(2b + ζ ) (β − α)
.
Some straightforward (but tedious) calculations enable us to write
I
ζ
2
(t) − I0
2 (t) =
ζtβ−α−1
b2
θ(1−β)/α,2(t) +
tβ−2α−1
b2
×
σ
ζu(1−β)/α exp[u1/α(1 + 4τ/t)](−u2
tα + u(2b + ζ )t2α − b(b + ζ )t3α )
[(u − (b + ζ )tα )exp[u1/ατ/t] − btα][(u − btα )exp[u1/ατ/t] − btα]
+
ζu(1−β)/α exp[u1/α(1 + 3τ/t)](2ubt2α − b(2b + ζ )t3α )
[(u − (b + ζ )tα )exp[u1/ατ/t] − btα][(u − btα )exp[u1/ατ/t] − btα]
du .
Utilizing similar estimates as in the proof of Proposition 5, we arrive at
|I
ζ
2
(t) − I0
2 (t)| ≤ |ζ|Ktβ−α−1
where K > 0 is a suitable real constant (independent of t and ζ). Further, continuity of L at ζ = 0 yields that for every ε
> 0 there exists χ(ε) > 0 such that if |ζ| < χ(ε) then |L(ζ ) − L(0)| < ε/3. Similarly, (49) implies that for every ε > 0 and
every ζ there exists N(ε, ζ) > 0 such that if t > N(ε, ζ) then |t1+α−β|I
ζ
2
(t)| − L(ζ )| < ε/3.
Employing the above stated properties, we can conclude that (49)1 holds also for a = b = 0 and α = β, i.e. for every ε >
0 there exists N(ε) > 0 such that if t > N(ε) then |t1+α−β|I0
2
(t)| − L(0)| < ε. Indeed, it holds
|t1+α−β|I0
2 (t)| − L(0)| ≤ |t1+α−β|I
ζ
2
(t)| − L(ζ )|
+ t1+α−β|I
ζ
2
(t) − I0
2 (t)| + |L(ζ ) − L(0)| ≤ ε/3 + ζK + ε/3 < ε
where ζ is chosen such that |ζ|K < ε/3.
II-2-c. Finally, we discuss the case a = b = 0 and α = β. Applying (48) with z = btα(exp[u1/ατ/t] + 1) and ξ =
u exp[u1/ατ/t] to (47), we obtain
I2(t) =
tα−1
2παi σ
−u1/α−1
exp[u1/α(1 + τ/t)]
btα(exp[u1/ατ/t] + 1)
−
u1/α exp[u1/α(1 + 2τ/t)]
b2t2α(exp[u1/ατ/t] + 1)2
+
u1/α+1
exp[u1/α(1 + 3τ/t)]
b2t2α(exp[u1/ατ/t] + 1)2((u − btα )exp[u1/ατ/t] − btα )
du .
Estimates employed in the proof of Proposition 5 yield
I2(t) =
−t−1
2παbi
G(t) + O(t−α−1
) + O(t−2α−1
) as t → ∞
where G, utilizing the change of variable u1/α = p, is given by
G(t) =
σ
u1/α−1
exp[u1/α(1 + τ/t)]
exp[u1/ατ/t] + 1
du = α
γ (ε,π/2+δ)
exp[p(1 + τ/t)]
exp[pτ/t] + 1
dp.
We can see that the integrand of the second expression is a function analytical in the complex plane, hence integral of this
function taken over any closed contour is zero. Utilizing this property, we can write
0 = lim
R→∞ (ε,δ,R)
exp[p(1 + τ/t)]
exp[pτ/t] + 1
dp =
G(t)
α
+ lim
R→∞
GR(t)
with the contour (ε, δ, R) depicted on Fig. 4. GR can be estimated as
|GR(t)| =
3π/2−δ
π/2+δ
exp[R(1 + τ/t)(cos(ϕ) + i sin(ϕ))]
exp[Rτ/t(cos(ϕ) + i sin(ϕ))] + 1
Ri exp[iϕ]dϕ
J. ˇCermák et al. / Applied Mathematics and Computation 298 (2017) 336–350 349
Fig. 4. The oriented closed contour (ε, δ, R) in the complex plane.
≤
R
2
exp[−R(1 + τ/t)sin(δ)](π − 2δ).
Since GR(t) → 0 as R → ∞ for any t > 0, we arrive at G(t) ≡ 0. Thus, it holds
I2(t) = O(t−α−1
) as t → ∞ .
III. Combining all the obtained results, we get Lemma 2 (i), (iii) and (iv). In particular, if all the roots of (12) have negative
real parts, then algebraic decay becomes dominant over the exponential with negative argument.
If (12) has the zero root, we have Ra,b,τ
α,β (t) = O(tβ−1) as t → ∞ (note that I1 equals zero) which corresponds to
Lemma 2 (ii) for β < 1. In the case β = 1, we can rewrite (9) as
Ra,b,τ
α,1
(t) = L−1 1
s
−
bs−1
(1 − exp[−τs])
sα + b(1 − exp[−τs])
(t) = 1 − b
t
t−τ
Ra,b,τ
α,α (ξ)dξ
= 1 − bτRa,b,τ
α,α (t − c(t)) = 1 + O(tα−1
) ∼ 1 as t → ∞
(0 ≤ c(t) ≤ τ) which concludes the proof of Lemma 2 (ii).
Remark 5. We note that our technique enables even a stronger formulation of Lemma 2 (i). In particular, if a = b then
Ra,b,τ
α,α (t) ∼ t−α−1 as t → ∞.
To complete the proof of Lemma 1, we consider (10) implying that every solution y of (1) consists of terms
φ(0)Ra,b,τ
α,1
(t) and b
0
−τ
Ra,b,τ
α,α (t − τ − u)h(t − τ − u)φ(u)du .
Let all the roots of (12) have negative real parts. By Lemma 2 (i), we have Ra,b,τ
α,1
(t) ∼ t−α and Ra,b,τ
α,α (t) = O(t−α−1) as
t → ∞ as t → ∞. For t > τ, we get
0
−τ
Ra,b,τ
α,α (t − τ − u)φ(u)du ≤ sup
t−τ≤u≤t
|Ra,b,τ
α,α (u)|
0
−τ
|φ(u)|du = O(t−α−1
)
as t → ∞. Thus, we arrive at (33)1 for φ(0) = 0 and (33)2 for φ(0) = 0.
350 J. ˇCermák et al. / Applied Mathematics and Computation 298 (2017) 336–350
Analogously, Lemma 2 (ii), (iii) and (iv) implies (34), (35) and (36), respectively. Moreover, the existence of a constant
solution of (1) in the case a + b = 0 (when the zero root of (12) occurs) can be checked via a direct substitution into (1).
Also, the property (36)1 or (36)2 occurs if the root of (12) with the maximal real part is double or single, respectively.
6. Concluding remarks
In this paper, we have discussed stability and asymptotic behaviour of a linear prototype of fractional delay differential
equations. The main results (Theorems 4 and 5) have formulated two types of necessary and sufficient conditions for stability
and asymptotic stability of the zero solution of (1), including asymptotic formulae for its solutions. The derived stability
conditions confirm expectations on positive influence of fractional-order derivative 0 < α < 1 on stability properties of
(1) compared with the classical case (2) when α = 1. The stability region for (1) is wider than that for (2) and, moreover,
the zero solution of (1) is stable for all couples (a, b) lying on the stability boundary (this property does not hold in the
case of (2) when the zero solution is not stable at the cusp point of the stability boundary). Another distinction consists in
the form of decay rate of solutions of (1) in the asymptotically stable case, which is not exponential, but algebraic.
Following the topic of this paper, we outline some next possible research directions. The most preferable direction is an
extension of our results to the vector case when (1) becomes
Dαy(t) = A y(t) + B y(t − τ ), t > 0 (50)
with d × d real matrices A, B. The knowledge od stability conditions for (50) is important from a theoretical as well as
practical view ((50) describes, among others, the fractional-order time delay state space model of PDα control of Newcastle
robot, see [16]). It should be noted that the problem of necessary and sufficient conditions for (asymptotic) stability of the
zero solution of (50) seems to be a very complicated matter which is not answered neither in the integer-order case α = 1
(for a particular result on this problem we refer to [13]). Also, these stability investigations can be extended to other types
of fractional equations involving more general types of delays (see, e.g. [19,22,23]), to fractional evolution equations (see,
e.g. [24,26]), or to basic numerical discretizations of (1). From this viewpoint, we hope that results of this paper might be a
starting point for numerical stability investigations of fractional delay differential equations.
Acknowledgements
The research was supported by the project FSI–S–14–2290 of Brno University of Technology.
The authors are grateful to a referee for reading this paper and his/her recommendations which helped to improve its
content.
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Appendix F
Paper on lower-order complex
two-term FDDE [11] (CNSNS, 2019)
The paper [11] (co-author: J. Čermák; my author’s share 50 %) was a reaction on
our growing interest in the phenomenon of stability switches. We decided to study a
planar FDDS with three real entries, which can be transformed into a fractional generalization
of the first-order delay differential equation with an imaginary coefficient
by an undelayed term. This type of classical system is known to switch stability on
and off with increasing time delay.
In this paper, we conducted a detailed analysis of stability switching, including
conditions for its appearance, number, and exact calculations of stability switches.
We discovered rich behaviour not present in the original first-order equation, such
as different switching patterns starting from instability or stability for small delays
and the presence of a delay-independent stability region.
The biggest challenge was the derivation of explicit form of the stability boundary,
which is no longer given by a continuous parametric curve but by a union of
segments from infinitely many parametric curves. Additionally, we explored how the
stability boundary transitions with increasing order of the derivative, from a union
of transcendental curves (for derivative orders less than one) to the known shape for
first-order equations.
– 133 –
Commun Nonlinear Sci Numer Simulat 79 (2019) 104888
Contents lists available at ScienceDirect
Commun Nonlinear Sci Numer Simulat
journal homepage: www.elsevier.com/locate/cnsns
Research paper
Delay-dependent stability switches in fractional differential
equations
Jan ˇCermák, Tomáš Kisela∗
Institute of Mathematics, Brno University of Technology, Technická 2, Brno CZ-616 69, Czech Republic
a r t i c l e i n f o
Article history:
Received 15 December 2018
Revised 24 April 2019
Accepted 20 June 2019
Available online 28 June 2019
MSC:
34K37
34K20
34K25
93D15
Keywords:
Fractional delay differential equation
Stability switch
Asymptotic behaviour
Stabilization
a b s t r a c t
This paper discusses stability properties of a linear fractional delay differential system involving
both delayed as well as non-delayed terms. As a main result, the explicit stability
dependence on a changing time delay is described, including conditions for the appearance,
number and exact calculations of stability switches for this system when its stability
property turns into instability and vice versa in view of a monotonically increasing lag.
Some supporting asymptotic results are stated as well. The proof technique is based on
analysis of the generalized delay exponential function of the Mittag-Leffler type combined
with D-decomposition method. The obtained results are illustrated via a fractional LotkaVolterra
population model and applied to a stabilization problem of the control theory.
© 2019 Elsevier B.V. All rights reserved.
1. Introduction
The usefulness of delay and fractional differential equations in mathematical modeling of various phenomena is well
known and described in many works. The presence of time delay parameters is connected especially with systems not capable
to react promptly after receiving an information (typically in chemical processes, electric and high-speed communication
networks, transportation phenomena and other related areas, see, e.g., [1]). Similarly, the involvement of non-integer derivative
orders turned out to be useful especially in models having hereditary properties (typically in viscoelasticity, acoustics,
diffusion, transmission lines and other problems, see, e.g., [2–5]).
Differential models comprising both a time delay as well as a non-integer derivative order found their applications especially
in various problems of control theory, robotics and related disciplines (see, e.g., [6–9]). In particular, both these
parameters can be viewed as control parameters in various stabilization and synchronization issues which makes stability
analysis of these equations very important. Although stability theory for delay differential equations and fractional differential
equations is well established, the corresponding unifying theory for fractional delay differential equations is still at the
beginning.
Stability dependence on some of changing system parameters belongs among the key qualitative properties of studied
systems. It is well known that the integer-order linear delay dynamical systems may repeatedly change their stability into
∗
Corresponding author.
E-mail addresses: cermak.j@fme.vutbr.cz (J. ˇCermák), kisela@fme.vutbr.cz (T. Kisela).
https://doi.org/10.1016/j.cnsns.2019.104888
1007-5704/© 2019 Elsevier B.V. All rights reserved.
2 J. ˇCermák and T. Kisela / Commun Nonlinear Sci Numer Simulat 79 (2019) 104888
instability and vice versa in view of a monotonically increasing time delay. This specific phenomenon was described, e.g., in
the papers [10–12] where explicit values of these stability switches were described in terms of remaining system parameters
(for switching laws designed to maintain stability of delayed switched nonlinear systems with stable and unstable modes see
also [13]). The existence of stability switches indicates that the involvement of a delay parameter into appropriate dynamical
system can result either into stabilizing or destabilizing effects depending on a concrete value of a delay. In the fractionalorder
case, the structure of stability dependence of linear dynamical systems on continuously changing derivative order is
more simple: the corresponding stability area becomes larger in view of a decreasing derivative order (see [14]). All these
facts play an important role in stabilization and synchronization problems connected with delayed feedback controls of
dynamical systems (see, e.g., [15–17]). A general theoretical framework related to various dynamical models was presented,
e.g., in [18,19].
The stability switches problem for fractional delay differential equations with delay-dependent coefficients was discussed
in [20,21]. While the first paper primarily investigated the stability intervals for such equations, in the latter one, using general
formulae based on the Jacobi determinant of some auxiliary functions, the occurrence of stability switches was observed
for studied fractional equations with delay-dependent as well as delay-independent coefficients. Despite of interesting results
involved in these papers, several important issues (concerning especially a priori conditions for the appearance, number
and exact values of stability switches) remained open.
Our main goal is to follow the above commented research and perform a detailed analysis of this phenomenon for a
basic fractional delay differential system admitting an arbitrary (finite) number of stability switches depending on values of
entry coefficients and a derivative order. Doing this, the paper is structured as follows. In Sections 2 and 3, we precise a
mathematical side of the problem, including related notions and tools. In particular, we derive the characteristic equation
associated to the studied system. Section 4 analyzes the location of characteristic roots which is a crucial step in related
stability investigations that are summarized in Section 5. We formulate here stability criteria for the studied fractional delay
differential system, providing a simple algorithm how to reveal the occurrence of stability switches, their number and exact
values. These values are derived explicitly in terms of system coefficients and a derivative order. In Section 6, we outline an
applicability of our results in stability analysis of equilibria of fractional delay dynamical systems and control theory. Some
final remarks conclude the paper.
2. A brief mathematical background
The fractional differential system
Dαx(t) = Ax(t) + Bx(t − τ ), (2.1)
where 0 < α < 1 is a real number, the symbol Dα means the Caputo derivative operator (its introduction is recalled below), A,
B are real 2 × 2 matrices and τ is a positive real time delay, provides a mathematical model with large application potential,
especially in control theory. In particular, (2.1) appears in stabilization of equilibria of fractional dynamical systems via
delayed feedback controls (A usually serves as the Jacobi matrix of an uncontrolled system and B is the gain matrix). For its
other use, we refer, e.g., to [8].
However, stability analysis of (2.1) represents a problem of a considerable difficulty. More precisely, if A, B are planar
matrices with general real entries aij, bij, respectively, then easily applicable stability conditions are missing even in the
integer-order case. Indeed, if α = 1, then the classical result says that the zero solution to (2.1) with α = 1 is asymptotically
stable if and only if all roots of the characteristic quasi-polynomial
F(s; A, B, τ ) ≡ det[sI − A − B exp(−sτ )]
= s2
− (tr A)s + det A + (−(tr B)s + a11b22 + a22b11 − a12b21 − a21b12)exp(−sτ ) + det B exp(−2sτ ) (2.2)
have negative real parts. Although there are several methods how to analyze this theoretical condition, necessary and sufficient
stability conditions for (2.1) with α = 1, given explicitly in terms of entry coefficients, are known only in very particular
cases. Perhaps, the best analyzed case is that when a factorization of F(s; A, B, τ) is possible, i.e. when A, B are
commutative (or, slightly more generally, simultaneously triangularizable). For relevant results on this topic we refer to
[22–25].
In the fractional-order case, an analogue of (2.2) is described and partially analyzed also in many special cases (for basic
results in this direction, see, e.g., [26]). However, stability conditions expressed explicitly in terms of entry coefficients are
still very rare. From a short history of searching for such stability conditions for (2.1), we refer to results on some of its basic
particular cases derived in [27] (the case A = 0, B ∈ R), [28] (the case A = 0, B ∈ Rd, d ∈ Z+) and [29–31] (the case A, B ∈ R).
In this paper, we follow the research and investigate stability properties of a planar fractional delay system
Dαx1(t) = ax1(t − τ ) + bx2(t)
Dαx2(t) = cx1(t) + ax2(t − τ ) (2.3)
with real entries a, b, c (b, c = 0) and a positive real delay τ. The conventional first-order pattern for (2.3) (with α = 1)
was studied in [24] where necessary and sufficient stability conditions explicit with respect to delay τ were derived. In
particular, the repeated switches between stability and instability with respect to changing time delay were observed and
J. ˇCermák and T. Kisela / Commun Nonlinear Sci Numer Simulat 79 (2019) 104888 3
expressed explicitly in terms of parameters a, b, c. Thus, (2.3) with α = 1 turned out to be the simplest conventional time
delay system enabling the occurrence of this phenomenon.
Therefore, we wish to extend stability analysis from [24] to its fractional analogue (2.3). As it might be expected, this
problem provides a considerably more difficult task. We show that stability switches with respect to changing time delay
can occur also in (2.3) and describe conditions for their appearance and number, including their exact calculations in terms
of a, b, c and α. Also, a supplemental information on algebraic decay rate of the solutions will be added.
3. Preliminary results
In this section, we recall some basic notions and formulate a theoretical basis related to stability analysis of (2.3). First,
we recall the fractional derivative operator Dα, 0 < α < 1 that is understood here in the Caputo sense. For a given real
function f, it is defined via the fractional integral
D−ν f (t) =
t
0
(t − ξ)ν−1
(ν)
f (ξ)dξ, ν > 0, t > 0
as
Dα f (t) = D−(1−α)
f (t), 0 < α < 1, t > 0 .
Further, we put D0 f (t) = f (t) (for more details on basics of fractional calculus theory we refer, e.g., to [4,32]).
The main analytical technique used in stability investigations of linear fractional differential equations is based on the
Laplace transform. For a given real function f, it is introduced as
L(f (t))(s) =
∞
0
f (t)exp (−st)dt , s ∈ D(f )
where D(f ) ⊂ C contains all complex s such that the integral converges. Employing this approach, the Laplace transform of
solution (x1, x2) to (2.3) can be expressed as
L
x1(t)
x2(t)
(s) = M−1
(s) · sα−1 φ1
φ2
(0) + a exp (−sτ )I
0
−τ
exp (−su)
φ1
φ2
(u)du (3.1)
where
M(s) =
sα − a exp (−sτ ) −b
−c sα − a exp (−sτ )
,
I is the planar identity matrix and φ1, φ2 are the associated initial functions that are piecewise continuous on [−τ, 0].
Obviously, (3.1) is well posed if and only if M(s) is regular, i.e. it holds
det M(s) = (sα − a exp (−sτ ))2
− bc = sα − bc − a exp (−sτ ) sα + bc − a exp (−sτ ) = 0 . (3.2)
Then the inverse of M(s) can be rewritten via the Jordan canonical form as
M−1
(s) =
1
2
b
c
− b
c
1 1
(sα −
√
bc − a exp (−sτ ))−1
0
0 (sα +
√
bc − a exp (−sτ ))−1
c
b
1
− c
b
1
.
Utilizing methods of the inverse Laplace transform, we can write the components x1, x2 of the solution to (2.3) in the
form
x1(t) =
φ1(0) − b
c
φ2(0)
2
R−
√
bc,a,τ
α,1
(t) +
φ1(0) + b
c
φ2(0)
2
R
√
bc,a,τ
α,1
(t)
+ a
0
−τ
φ1(u) − b
c
φ2(u)
2
R−
√
bc,a,τ
α,α (t − τ − u) +
φ1(u) + b
c
φ2(u)
2
R
√
bc,a,τ
α,α (t − τ − u) h(t − τ − u)du ,
(3.3)
x2(t) =
φ2(0) − c
b
φ1(0)
2
R−
√
bc,a,τ
α,1
(t) +
φ2(0) + c
b
φ1(0)
2
R
√
bc,a,τ
α,1
(t)
+ a
0
−τ
φ2(u) − c
b
φ1(u)
2
R−
√
bc,a,τ
α,α (t − τ − u) −
φ2(u) + c
b
φ1(u)
2
R
√
bc,a,τ
α,α (t − τ − u) h(t − τ − u)du
(3.4)
4 J. ˇCermák and T. Kisela / Commun Nonlinear Sci Numer Simulat 79 (2019) 104888
for t > 0, where h is the Heaviside step function and Rv,w,τ
α,β is a generalized delay exponential function of the Mittag-Leffler
type introduced in [31] as
Rv,w,τ
α,β (t) = L−1 sα−β
sα − v − w exp[−sτ]
(t), α, β, τ ∈ R+
and v, w ∈ C . (3.5)
Thus, the relations (3.3) and (3.4) enable us to study asymptotic properties of x1, x2 through asymptotics of Rv,w,τ
α,β with
v = ±
√
bc, w = a and β ∈ {α, 1}.
Now we discuss next procedures that depend on the sign of bc. If bc > 0, then both v, w are real numbers, hence the
behaviour of Rv,w,τ
α,β is covered by some parts of [31]. In particular, the results relevant for our purposes can be summarized
in the following
Lemma 1. Let 0 < α < 1, 0 < β ≤ 1, τ > 0, v and w be real numbers. We denote
τ∗
(v, w) =
(2 − α)π/2 + arcsin(v/w sin(απ/2))
w2 − v2 sin
2
(απ/2) + vcos(απ/2)
1/α
. (3.6)
Then Rv,w,τ
α,β tends to zero as t → ∞ if and only if
v ≤ w < −v and τ > 0, or |v| + w < 0 and τ < τ∗
(v, w).
Using this conclusion, we can describe asymptotic properties of (3.3) and (3.4) when bc > 0.
If bc < 0, then v remains a real number, but w is purely imaginary, hence the above mentioned result cannot cover
this case. Thus, the next section is devoted solely to study of the case bc < 0. In particular, we focus on root analysis of
characteristic equations implied by (3.2) (and consequently (3.5)) in the form
Q1(s) ≡ sα − ip − q exp (−sτ ) = 0 and Q2(s) ≡ sα + ip − q exp (−sτ ) = 0 (3.7)
where p =
√
−bc = 0, q = a are real numbers.
4. Analysis of characteristic roots
As recalled above, the question of stability of linear autonomous differential systems, both integer-order as well as
fractional-order, is closely related to the location of all characteristic roots to the left of the imaginary axis. As we shall
confirm later, (2.3) is not an exception to this rule. Thus, this section is devoted to a study of conditions guaranteeing that
all roots of (3.7) have negative real parts.
We start with a survey of several generic properties of roots of (3.7).
Proposition 1. Let 0 < α < 1 and p, q, τ be real numbers such that p = 0 and τ > 0.
(i) If s is a root of Q1, then its complex conjugate ¯s is a root of Q2.
(ii) Q1 and Q2 have no non-negative real roots.
(iii) Let 0 < ω < π be arbitrary. Then Q1 and Q2 have only a finite number of roots s such that |arg(s)| ≤ ω.
(iv) All roots of Q1 and Q2 have multiplicity one or two.
Proof. The property (i) follows immediately by splitting Q1 and Q2 into real and imaginary parts. Its validity enables us to
prove the remaining properties only with respect to Q1.
Let s0 be a non-negative real root of Q1. Then the expression sα
0
− q exp (−s0τ ) has to be purely imaginary (and non-zero)
which is a contradiction. Hence, the property (ii) is valid.
The technique how to verify (iii) is the same as that used in the proof of [31, Proposition 2 (iii)].
Finally, let s0 be a root of Q1 with multiplicity larger than two. Then
Q1(s0) = αsα−1
0 + qτ exp (−s0τ ) = 0 and Q1 (s0) = α(α − 1)sα−2
0 − qτ2
exp (−s0τ ) = 0 .
This implies that s0 is a positive real number, namely s0 = (1 − α)/τ, which contradicts (ii). Thus, the property (iv) holds as
well.
Due to Proposition 1 (i), real parts of roots of Q1 and Q2 are identical, hence it is enough to consider Q1 only. To (formally)
simplify later discussions, we introduce a substitution z = sτ enabling us to study ˜Q1 instead of Q1 where
˜Q1(z) ≡ zα − iτα p − ταq exp(−z) = 0 , (4.1)
J. ˇCermák and T. Kisela / Commun Nonlinear Sci Numer Simulat 79 (2019) 104888 5
0 < α < 1, τ > 0, p = 0 and q still being real numbers. We wish to find an explicit characterization of a property guaranteeing
that all roots of ˜Q1 are located in the left half-part of the complex plane. For this purpose, we first use the D-decomposition
method. Let z = ρ exp(iϕ) for suitable reals ρ > 0 and −π < ϕ ≤ π. Then z is the root of ˜Q1 if and only if
ρα cos αϕ − ταq exp (−ρ cos ϕ)cos(ρ sin ϕ) = 0
ρα sin αϕ − τα p + ταq exp (−ρ cos ϕ)sin(ρ sin ϕ) = 0 .
For ϕ = ±π/2, we get the boundary locus curves in a form
τα p = ±
ρα sin(ρ + απ/2)
cos ρ
ταq =
ρα cos απ/2
cos ρ
, ρ > 0 , ρ =
π
2
+ jπ , ρ + απ/2 = jπ , j = 0, 1, 2, . . . . (4.2)
Consequently, for any couple (ταp, ταq) given by (4.2), (4.1) has a purely imaginary root. Moreover, it holds
Proposition 2. Let 0 < α < 1, τ > 0 and p = 0 be fixed real numbers and let q be a free real number such that (ταp, ταq) is given
by (4.2) for a suitable ρ > 0 and a suitable sign combination. If |q| increases, then a new root of (4.1) having a positive real part
appears.
Proof. For any purely imaginary z, there exists its neighborhood where ˜Q1 is an analytic function and its roots are continuously
depending on parameters p, q, τ, α (in fact, ˜Q1 is analytic for all points apart from non-positive reals). We show that if
a couple (ταp, ταq) from (4.2) is considered and |q| increases, then the real part of the corresponding purely imaginary root
becomes positive. Indeed, if we consider z as a function of p, q given implicitely by (4.1) and differentiate z with respect to
q, we get
dz
dq
=
τα exp(−z)
αzα−1 + ταq exp(−z)
=
zα − iτα p
q(αzα−1 + zα − iτα p)
,
which, in the case of purely imaginary root z = iρ (ρ > 0), yields
dz
dq
z=+iρ =
(iρ)α − iτα p
q(α(iρ)α−1 + (iρ)α − iτα p)
=
1
q
ρα cos(απ/2) + i(ρα sin(απ/2) − τα p)
αρα−1 cos((α − 1)π/2) + ρα cos(απ/2) + i(αρα−1 sin((α − 1)π/2) + ρα sin(απ/2) − τα p)
=
1
q
a1 + ia2
a3 + ia4
=
1
q
(a1a3 + a2a4) + i(a2a3 − a1a4)
a2
3
+ a2
4
where a1 = ρα cos(απ/2), a2 = ρα sin(απ/2) − τα p, a3 = αρα−1 cos((α − 1)π/2) + ρα cos(απ/2) and a4 = αρα−1 sin((α −
1)π/2) + ρα sin(απ/2) − τα p. Considering the derivative of a real part of the root with respect to q, we obtain
dR(z)
dq
z=+iρ = R
dz
dq
z=+iρ =
1
q
a1a3 + a2a4
a2
3
+ a2
4
where
a1a3 + a2a4 = (ρα − τα p)2
+ 2ρατα p(1 − sin(απ/2)) + αρα−1
τα pcos(απ/2) > 0 for p > 0 .
Due to Proposition 1 (i), the real parts of roots of (4.1) are not depending on the sign of p, so we can extend this result for
all real p to get
sgn
dR(z)
dq
z=+iρ = sgn(q). (4.3)
Analogously, we can extend (4.3) also for z = −iρ (ρ > 0). Thus, whenever we are crossing any curve from the system
(4.2) with α, τ, p fixed and |q| increasing, a new root of (4.1) with a positive real part actually appears.
The next assertion is useful in calculations of cusp points where relevant parts of the boundary locus system intersect
each other.
Proposition 3. Let 0 < α < 1. Then, for any j = 0, 1, 2, . . . , the equation
−
sin(θ + απ/2)
sin(θ − απ/2)
=
(2 j + 3 − α)π
θ + (j + 1/2 − α/2)π
− 1 α (4.4)
has a unique root θ∗
j
such that θ∗
j
∈ (0, απ/2). Moreover, it holds θ∗
j
> θ∗
j+1
for all j = 0, 1, 2, . . . and θ∗
j
→ 0+ as j → ∞.
Proof. Put
gj(θ) =
sin(θ + απ/2)
sin(θ − απ/2)
+
(2 j + 3 − α)π
θ + (j + 1/2 − α/2)π
− 1 α .
6 J. ˇCermák and T. Kisela / Commun Nonlinear Sci Numer Simulat 79 (2019) 104888
We show that gj(θ) has a unique root θ∗
j
in (0, απ/2). Since
d
dθ
sin(θ + απ/2)
sin(θ − απ/2)
= −
sin(απ/2)
sin
2
(θ − απ/2)
< 0 ,
gj(θ) is decreasing in (0, απ/2). Moreover,
gj(θ) → 1 +
4
2 j + 1 − α
α
− 1 > 0 as θ → 0+
and
gj(θ) → −∞ as θ → απ/2 −
.
The uniqueness of θ∗
j
is now ensured by continuity of gj(θ) in (0, απ/2).
Moreover, a direct calculation shows that gj(θ) > gj+1(θ) for all θ ∈ (0, απ/2) and j = 0, 1, 2, . . . . This fact, along with
continuity and decrease of gj(θ) with respect to θ, implies the property θ∗
j
> θ∗
j+1
. Letting j → ∞, gj(θ) is reduced to g∞(θ) =
sin(θ + απ/2)/ sin(θ − απ/2) + 1 with a root θ∗
∞ = 0 which concludes the proof.
To describe the boundary of the set of all (ταp, ταq) such that all roots of (4.1) have negative real parts, we consider
two piecewise smooth curves + and − defined in the (ταp, ταq)-plane as follows: Let θ∗
j
∈ (0, απ/2) be the unique root
of (4.4) for j = 0, 1, 2, . . . . We put
ρ∗
j =
(2 j + 1 − α)π
2
+ θ∗
j , ρ∗∗
j =
(2 j + 1 − α)π
2
− θ∗
j−2, j = 0, 1, 2, . . .
where we define
θ∗
−2 =
(α − 1)π
2
, θ∗
−1 =
π
2
(this leads to ρ∗∗
0
= 0 and ρ∗∗
1
= (2 − α)π/2). Further, for any j = 0, 1, 2, . . . , we introduce the system of curves j in the
(ταp, ταq)-plane given via a parametric form
τα p = ±
ρα sin(ρ + απ/2)
cos(ρ)
ταq =
ρα cos(απ/2)
cos(ρ)
, ρ∗∗
j ≤ ρ ≤ ρ∗
j (4.5)
(in the first relation, both the sign cases have to be considered, and thus any curve j has two branches symmetric with
respect to ταq-axis). Finally, we define + = ∪∞
j=0 2 j and − = ∪∞
j=0 2 j+1.
Using this notation, we have
Lemma 2. Let 0 < α < 1, τ > 0, p = 0 and q be real numbers. All roots of (4.1) have negative real parts if and only if the couple
(ταp, ταq) is located inside the area bounded by + from above and by − from below.
Proof. For q = 0 and p = 0, (4.1) is reduced to the equation zα − iτα p = 0 having roots with a negative real part only (we
consider the principal branches). Then Proposition 2 implies that all roots of (4.1) have negative real parts if and only if the
couple (ταp, ταq) lies in a region containing the points ταq = 0 and bounded by appropriate boundary locus curves. We
show that these curves are just + and −.
As already noted above, the boundary locus consists of countably many continuous branches of (4.2). For any j =
0, 1, 2, . . . , we put
fj(ρ) =
cos(απ/2)
sin(ρ + απ/2)
, (j − 1)π + π/2 < ρ < jπ + π/2 , ρ + απ/2 = jπ . (4.6)
Using this notation we can describe the ratio q/|p|, where p, q are given by (4.2), as
q
|p|
= fj(ρ), (j − 1)π + π/2 < ρ < jπ + π/2 , ρ + απ/2 = jπ .
Obviously, fj has an extremal value (−1)j cos(απ/2) occurring when ρ = (j + (1 − α)/2)π ∈ [ρ∗∗
j
, ρ∗
j
] (indeed, d fj/dρ =
−q/|p|cot(ρ + απ/2)). In particular, the minimal value of the ratio q/|p| related to + equals cos (απ/2) for all its branches
(the case when j is even), while the maximal value of q/|p| related to − equals − cos(απ/2) for all its branches (the case
when j is odd). By Proposition 2, it is enough, for each of the branches, to find intersections with boundary locus curves
closest to the corresponding minimum or maximum.
We consider an arbitrary even j ≥ 2 and calculate the intersection between two adjacent branches of boundary locus
curves from the system (4.2). Let
sin(ρ1 + απ/2) = sin(ρ2 + απ/2) (4.7)
J. ˇCermák and T. Kisela / Commun Nonlinear Sci Numer Simulat 79 (2019) 104888 7
Fig. 1. A classical result for the stability region (α = 1).
Fig. 2. The stability region for α = 0.95.
and
ρα
1
cos(ρ1)
=
ρα
2
cos(ρ2)
(4.8)
for a suitable (j − 1)π + π/2 < ρ1 < jπ + π/2 and (j + 1)π + π/2 < ρ2 < (j + 2)π + π/2 (note that for j = 0 this interval
is replaced by 0 < ρ1 < π/2). Then (4.7) implies either ρ1 + 2π = ρ2 (that, however, contradicts (4.8)), or ρ1 + ρ2 = (2 j + 3 −
α)π. Substituting this into (4.8) we have
ρα
1
cos(ρ1)
=
((2 j + 3 − α)π − ρ1)α
cos((2 j + 3 − α)π − ρ1)
, i.e. −
cos(ρ1 + απ)
cos(ρ1)
=
(2 j + 3 − α)π
ρ1
− 1 α .
Denoting θ = ρ1 − (j + 1/2 − α/2)π, Proposition 3 directly implies that this equation has a unique root in a given interval.
This root corresponds to the (unique) intersection of appropriate boundary locus curves. It is easy to check that relevant
parts of these curves form just the system of curves j. Employing the same technique, we can show that this intersection
is actually the closest one to the extreme (in the given direction), i.e. intersections with any non-adjacent branch occur for
ρ > ρ∗
j
or ρ < ρ∗∗
j
. The same argumentation is true also when j is odd.
Relevant parts of (ταp, ταq)-plane, where all roots of (4.1) have negative real parts, are depicted in Figs. 1–4 for various
values of α, including the first-order case discussed in [24] (these parts are referred to as stability regions in comments to
figures and are depicted in a grey color). Contrary to the first-order case (Fig. 1), any stability region for 0 < α < 1 (Figs. 2–4)
forms a set connected in the (ταp, ταq)-plane. Also, we can observe larger stability regions for decreasing α (more detailed
comments will be given in Remark 2).
8 J. ˇCermák and T. Kisela / Commun Nonlinear Sci Numer Simulat 79 (2019) 104888
Fig. 3. The stability region for α = 0.6.
Fig. 4. The stability region for α = 0.2.
For a given ratio |q/p|, we can give an analytical description of delays τ such that (τα p, ταq) ∈ + ∪ −. On this account,
we put
τ∗
i,κ (p, q) =
(i + (1 − α)/2)π − κ arccos(|p/q|cos(απ/2))
κ q2 − p2 cos2(απ/2) + |p|sin(απ/2) 1/α
, i = 0, 1, 2, . . . , κ = ±1 . (4.9)
If |q/p| ≥ cot(απ/2), then the symbol τ∗
0,1
is not defined.
Using this notation we have
Lemma 3. Let 0 < α < 1, τ > 0, p = 0 and q be real numbers such that |q/p| > cos (απ/2). Further, let θ∗
j
∈ (0, απ/2) be the
unique root of (4.4) for j = 0, 1, . . . . Then it holds:
(i) There exists a unique couple of consecutive non-negative integers m such that
cos(απ/2)
cos(θ∗
m)
≤
|q|
|p|
<
cos(απ/2)
cos(θ∗
m−2
)
, if m ≥ 2 , or
cos(απ/2)
cos(θ∗
m)
≤
|q|
|p|
, if m ∈ {0, 1}. (4.10)
(ii) If (τα p, ταq) ∈ + ∪ −, then τ = τ∗
i,κ (p, q) for a suitable i = 0, 1, . . . , m and κ = ±1 (see (4.9)). Here, i and m given by
(4.10) are even (odd) when q ≥ 0 (q < 0), respectively.
(iii) τ∗
i,1
≤ τ∗
i,−1
< τ∗
i+2,1
for all i = 0, 1, . . . , m − 2.
Proof. (i) Proposition 3 guarantees that θ∗
m > θ∗
m+2 are both lying in (0, απ/2) for all m = 0, 1, 2 . . . , which ensures the well
posedness of (4.10). Moreover, Proposition 3 implies that cos(απ/2)/ cos(θ∗
m) → cos(απ/2) as m → ∞ which, along with the
condition |q/p| > cos (απ/2), proves (i).
J. ˇCermák and T. Kisela / Commun Nonlinear Sci Numer Simulat 79 (2019) 104888 9
(ii) If we substitute the endpoints of j into (4.6), we get
fj(ρ∗
j ) =
cos(απ/2)
(−1)j cos(θ∗
j
)
, fj(ρ∗∗
j ) =
cos(απ/2)
(−1)j cos(θ∗
j−2
)
.
We note that the endpoints of j (corresponding to ρ∗
j
and ρ∗∗
j
) are actually cusp points of + or − (in the sequel, we call
them the j-th and the (j − 2)-nd cusp point, respectively; obviously, the cusp points are not defined when j ∈ {0, 1}). Even
values of j relate to + and odd j correspond to −. We point out that fj(ρ) is convex or concave for ρ lying in appropriate
intervals corresponding to + or −, respectively. The property (i) implies that (4.10) determines two couples of adjacent
cusp points (the m-th and the (m − 2)-nd) of + and −, respectively. Each of couples is characterized by the property that
tangents of two lines m, m−2 (m ≥ 2) starting from the origin and going through the corresponding cusp points define an
interval containing |q/p|.
Let (4.10)1 be valid. The above arguments yield that m contains a unique point of the given ratio |q/p| and every m−2 j
( j = 1, . . . (m − 1)/2) has two such points with a possible exception of 0. Indeed, since 0 starts from the origin with the
tangent equal to cot(απ/2), it has two such points only when |q/p| < cot(απ/2), otherwise it has only one. In the case of
(4.10)2, there is only one point of the given ratio |q/p| lying on 0 (provided |q/p| < cot(απ/2)) or 1 depending on the
sign of q. Thus, for |q/p| being fixed, all points (τα p, ταq) ∈ + ∪ − lie on branches j such that j ≤ m (j being even or odd
for + or −, respectively). Now we are going to derive the precise expressions for the values of τ characterizing these
points.
First, we consider a part of j (j ≥ 2 being an even integer) corresponding to the parameter range ρ ∈ (ρ∗∗
j
, (j + 1/2 −
α/2)π), i.e. the part of j corresponding to decreasing fj(ρ) (see the proof of Lemma 2). Then (4.6) yields
ρ + απ/2 = jπ + arcsin (|p|/q cos(απ/2)), i.e. ρ = (j + (1 − α)/2)π − arccos (|p|/q cos(απ/2)).
A substitution into (4.5)2 leads to
q =
[(j + (1 − α)/2)π − arccos(|p|/q cos(απ/2))]α cos(απ/2)
τα cos[(j + (1 − α)/2)π − arccos(|p|/q cos(απ/2))]
.
Since
q cos[(j + (1 − α)/2)π − arccos(|p|/q cos(απ/2))] = q2 − p2 cos2(απ/2) + |p|sin(απ/2) cos(απ/2)),
we have
τα =
[(j + (1 − α)/2)π − arccos (|p|/q cos(απ/2))]α
q2 − p2 cos2(απ/2) + |p|sin(απ/2)
.
Similarly, for a part of j (where j ≥ 2 is still even) with ρ ∈ ((j + 1/2 − α/2)π, ρ∗
j
), i.e. the part corresponding to increasing
fj(ρ), one gets
τα =
[(j + (1 − α)/2)π + arccos(|p|/q cos(απ/2))]α
− q2 − p2 cos2(απ/2) + |p|sin(απ/2)
.
That yields (4.9) for j ≥ 2 being even and κ = ±1, where positive or negative κ corresponds to decreasing or increasing
fj(ρ), respectively. If j = 0, then all the calculations are analogous up to the case |q/p| ≥ cot(απ/2) when computations for
ρ ∈ (0, (1 − α)π/2) leads to ρ < 0 (or τ < 0) which is a contradiction. Therefore, (4.9) is not defined for j = 0 and κ = 1 in
this case.
The technique can be directly extended for odd integers j ≥ 1, where κ = 1 and κ = −1 correspond to decreasing and
increasing |fj(ρ)|, respectively.
(iii) The last assertion of Lemma 3 follows from the above argumentation. E.g., if j is even, then τ = τ∗
j,1
(τ = τ∗
j,−1
)
corresponds to the point (ταp, ταq) lying on the decreasing (increasing) segment of j ⊂ +, respectively. The validity of
the inequalities now follows from a simple computational analysis of (4.5).
Remark 1. Lemma 3 plays a key role when finding the relation between the ratio |q/p| and cusp points of +, −. Another
significant value for |q/p| is the tangent cot(απ/2) of + at the origin. As it was already pointed out, lines m (m ≥ 0 being
even) connecting the origin and cusp points of + have decreasing tangents with respect to increasing index of given cusp
points (a similar comment is true also for −). It might be useful to discuss a relationship between the cusp points and the
above mentioned value cot(απ/2). On this account, we consider the first cusp point of + (i.e. the endpoint of 0). If we
equate tangents of + at the origin and a line 0, and supplied with the condition for the first cusp point (see the proof of
Lemma 2), we arrive at the system of two equations
cot(απ/2) =
cos(απ/2)
cos(θ)
−
sin(θ + απ/2)
sin(θ − απ/2)
=
(3 − α)π
θ + (1 − α)π/2
− 1 α (4.11)
10 J. ˇCermák and T. Kisela / Commun Nonlinear Sci Numer Simulat 79 (2019) 104888
for two unkowns 0 < α < 1 and 0 < θ < απ/2. It can be shown that (4.11) has a unique solution that can be numerically
calculated as
α∗
≈ 0.6150768144 and θ∗
0 ≈ 0.562788670 . (4.12)
Consequently, if α > α∗, then the tangent of + at the origin is larger than the tangent of 0 passing through the origin and
the first cusp point of +.
5. Main results
In this section, we summarize previous considerations to obtain stability criteria for (2.3) that are given in terms of entry
coefficients with respect to the sign of bc.
Theorem 1. Let 0 < α < 1, τ > 0 and a, b, c be real numbers such that bc > 0 and let τ∗ be defined by (3.6). The zero solution to
(2.3) is asymptotically stable if and only if
a + bc < 0 and τ < τ∗
( bc, a).
Proof. By (3.3), (3.4), the solution to (2.3) can be expressed in terms of the functions R±
√
bc,a,τ
α,α , R±
√
bc,a,τ
α,1
. More precisely,
the zero solution to (2.3) is asymptotically stable if and only if both R
√
bc,a,τ
α,α (t) and R−
√
bc,a,τ
α,α (t) tend to zero as t → ∞.
Although Lemma 1 admits two families of conditions guaranteeing this property, we can see that the first of them (corresponding
to a delay-independent part) cannot be satisfied simultaneously for both v =
√
bc and v = −
√
bc. A direct sign
analysis of (3.6) yields that τ∗(
√
bc, a) < τ∗(−
√
bc, a) for all a < 0 (positive values of a are excluded due to the condition
a +
√
bc < 0). The assertion now follows from Lemma 1.
Theorem 2. Let 0 < α < 1, τ > 0 and a, b, c be real numbers such that bc < 0, let α∗ be given by (4.12) and let θ∗
j
∈ (0, απ/2)
be the unique root of (4.4) for j = 0, 1, 2, . . . . Further, assuming |a|/
√
−bc ≥ cos(απ/2), let m ≥ 0 be an even integer (if a ≥ 0),
or an odd integer (if a < 0), uniquely determined by
cos(απ/2)
cos(θ∗
m)
≤
|a|
√
−bc
<
cos(απ/2)
cos(θ∗
m−2
)
, m ≥ 2 , or
cos(απ/2)
cos(θ∗
m)
≤
|a|
√
−bc
, m ∈ {0, 1}. (5.1)
The zero solution to (2.3) is asymptotically stable if and only if any of the following conditions holds:
|a|
√
−bc
< cos(απ/2); (5.2)
cos(απ/2) ≤
a
√
−bc
< cot(απ/2) and τ ∈
m/2−1
j=−1
τ∗
2 j,−1, τ∗
2 j+2,1 ; (5.3)
α > α∗
, cot(απ/2) ≤
a
√
−bc
<
cos(απ/2)
cos(θ∗
0
)
and τ ∈
m/2−1
j=0
τ∗
2 j,−1, τ∗
2 j+2,1 ; (5.4)
a
√
−bc
≤ − cos(απ/2) and τ ∈
(m−1)/2−1
j=−1
τ∗
2 j+1,−1, τ∗
2 j+3,1 , (5.5)
where τ∗
i,κ = 0 for negative integers i and τ∗
i,κ = τ∗
i,κ (
√
−bc, a) are given by (4.9) for non-negative integers i.
Remark 2. (i) The condition (5.2) describes the delay-independent asymptotic stability area for (2.3). While this area is
empty when α = 1 (see also [24]), it becomes non-empty for any 0 < α < 1. Moreover, an explicit dependence of this area on
a changing derivative order α is described via (5.2). The remaining conditions (5.3)–(5.5) capture delay-dependent asymptotic
stability areas for (2.3). It might be interesting to discuss their form with respect to α approaching 1. Similarly to (5.2),
the condition (5.3) becomes empty when α → 1. Further, in this limit case, (5.4) turns into the stability condition (1.6) of
[24], and (5.5) can be split into (1.4)–(1.5) of [24]. The same comment is true for the stability condition of Theorem 1 that
can be converted into the condition (1.3) of [24]. Thus, stability conclusions of Theorems 1 and 2 with α approaching 1 fully
agree with existing stability conditions for the corresponding integer-order system.
(ii) As observed in Figs. 1–4, stability regions become larger with decreasing α which can be utilized, e.g., in stabilization
or synchronization issues, or in design of fractional PID controllers which were studied, along with influence of α as a
control parameter, by many authors (see, e.g., [4,17]). If we formally put α = 0 in (2.3) and assume bc = 1, then we obtain,
after some technical rearrangements, a difference system with the continuous time in a form
x1(t) =
a
1 − bc
(x1(t − τ ) + bx2(t − τ ))
x2(t) =
a
1 − bc
(cx1(t − τ ) + x2(t − τ ))
J. ˇCermák and T. Kisela / Commun Nonlinear Sci Numer Simulat 79 (2019) 104888 11
Fig. 5. A classical result for the stability region (α = 1), see [24]; delay-independent instability above the upper dotted line, stability switching between
dotted lines, one-time stability loss below the lower dotted line.
whose stability properties depend on location of the eigenvalues of the system matrix inside the unit circle of the complex
plane. The use of the Schur–Cohn test (see, e.g., [33]) implies that this occurs if and only if a2 + bc < 1 (hence, the stability
boundaries depicted in Figs. 1–4 are simplified to the hyperbola when α = 0). This agrees with the conclusion of Theorem 2,
whose stability condition actually become a2 + bc < 1 when α is approaching zero.
(iii) The conditions (5.3)–(5.5) of Theorem 2 reveal that (2.3) actually provides (probably the simplest) fractional differential
system admitting the stability switches phenomenon. If a ≥ 0, then repeated stability switching (with more
than one change of stability properties) occurs when cos(απ/2) ≤ a/
√
−bc < cos(απ/2)/ cos(θ∗
0
), while for a < 0 when
cos(απ/2)/ cos(θ∗
1 ) > |a|/
√
−bc ≥ cos(απ/2). To determine an exact number of stability switches, nonlinear conditions
(4.4) and (5.1) have to be solved. We note that the uniqueness of appropriate solutions to (4.4) and (5.1) is guaranteed
by Proposition 3 and Lemma 3 (i) (with p =
√
−bc, q = a), respectively.
(iv) If |a|/
√
−bc = cos(απ/2), then τ∗
i,1
= τ∗
i,−1
for all i ≥ 0. Hence, the zero solution to (2.3) is asymptotically stable for
all τ > 0 except for infinitely (but countably) many values
τ = τ∗
i,1 = τ∗
i,−1 =
(i + (1 − α)/2)π
(−bc − a2)1/(2α)
with i being even for a ≥ 0 and odd for a < 0. At these delay values, a non-asymptotic stability appears.
(v) In the asymptotically stable case, any solution (x1, x2) to (2.3) is tending to the zero solution with an algebraic decay
rate
|xj(t)| ∼ t−α as t → ∞ , j = 1, 2
(the symbol ∼ means an asymptotic equivalence). This asymptotic property is commented at the end of the following proof.
(vi) A rich stability behaviour of (2.3) with bc < 0 (described in Theorem 2) is graphically illustrated in Figs. 5–8.
Proof. Throughout the proof, we keep the notation p =
√
−bc, q = a, hence p > 0 and q ∈ R. First, we prove that conditions
(5.2)–(5.5) are necessary and sufficient for all roots of Q1 and Q2 (defined by (3.7)) to have negative real parts. By
Proposition 1 (i) and a discussion following its proof, it is enough to consider ˜Q1 given by (4.1). By Lemma 2, it is sufficient
to prove that the conditions (5.2)–(5.5) capture just couples (ταp, ταq) located between + and −.
As it was shown in the proof of Lemma 3, all the couples (ταp, ταq) with |q|/p < cos (απ/2) lie between + and −,
which proves (5.2).
Now, let q ≥ 0. Then a position of (ταp, ταq) has to be evaluated only with respect to + (for q/p ≥ cos (απ/2)). Consider
a line starting from the origin with a tangent q/p. Lemma 3 implies that has either m or m + 1 intersections with +
where m is determined by (4.10) (and therefore by (5.1)). These intersections τ∗
i,κ are characterized by (4.9) for i = 0, 2, . . . , m
(i, m being even) and τ∗
i,1
≤ τ∗
i,−1
< τ∗
i+2,1
.
If cos(απ/2) ≤ q/p < cot(απ/2), then the line starts below 0, i.e. below +. Indeed, as indicated in the proof of
Lemma 3, + starts from the origin with a tangent equal to cot(απ/2) (i.e. it is lying in the first quadrant of (ταp, ταq)plane).
Each of intersections τ∗
i,κ (i = 0, 2, . . . , m) then describes a point where crosses (or in the case q/p = cos(απ/2)
only touches) + which implies (5.3). If q/p ≥ cot(απ/2), the line starts above 0, i.e. above +. Then Remark 1 implies
that intersects + only when α > α∗, and a similar argumentation as used above leads to (5.4). Finally, if q/p ≥ cot(απ/2)
and α ≤ α∗, or q/p ≥ cos(απ/2)/ cos(θ∗
0
) and α > α∗, then (ταp, ταq) is located above +.
12 J. ˇCermák and T. Kisela / Commun Nonlinear Sci Numer Simulat 79 (2019) 104888
Fig. 6. The stability region for α = 0.8 > α∗
; delay-independent instability above the upper dotted line, stability switching starting from instability between
upper dotted and dash-dotted lines, stability switching starting from stability between dash-dotted and upper dashed lines and between lower dashed and
dotted lines, delay-independent stability between dashed lines, one-time stability loss below the lower dotted line.
Fig. 7. The stability region for α = α∗
; delay-independent instability above the dash-dotted line, stability switching starting from stability between dashdotted
and upper dashed lines and between lower dashed and dotted lines, delay-independent stability between dashed lines, one-time stability loss below
the lower dotted line.
Fig. 8. The stability region for α = 0.4 < α∗
; delay-independent instability above the dash-dotted line, one-time stability loss between the dash-dotted and
upper dotted lines and below the lower dotted line, stability switching starting from stability between the upper dotted and dashed lines and between
lower dashed and dotted lines, delay-independent stability between dashed lines.
J. ˇCermák and T. Kisela / Commun Nonlinear Sci Numer Simulat 79 (2019) 104888 13
Let q < 0. Then a position of (ταp, ταq) has to be evaluated only with respect to − (for q/p ≤ − cos(απ/2)). We again
introduce as a line starting from the origin with a tangent q/p (i.e. lying in the fourth quadrant of (ταp, ταq)-plane).
(5.5) can be now proved by the same line of arguments as used for (5.3).
Thus we have shown that the conditions (5.2)–(5.5) are valid if and only if Q1 and Q2 have all roots with negative real
parts.
Analogously to the proof of Theorem 1, stability properties of (2.3) with bc < 0 are determined by appropriate asymptotic
properties of the functions R±i
√
−bc,a,τ
α,α , R±i
√
−bc,a,τ
α,1
. Based on characteristic root properties stated in Proposition 1 (ii)-(iv),
we can utilize the proof technique from [31] (supplied with some straightforward modifications) to show that R±i
√
−bc,a,τ
α,α (t)
and R±i
√
−bc,a,τ
α,1
(t) tend to zero as t → ∞ if and only if (3.7) has all roots with negative real parts. This proves the assertion
of Theorem 2. A note on algebraic decay of the solutions (see Remark 2 (iv)) follows from related considerations originating
from the technique used in [31]. It implies that, in the asymptotically stable case, R±i
√
−bc,a,τ
α,1
(t) ∼ t−α and R±i
√
−bc,a,τ
α,α (t) =
O(t−α−1) as t → ∞.
Remark 3. We note that the stability conditions of Theorems 1 and 2 can be reformulated for (2.1), where A is a planar
matrix having either two real eigenvalues of the form ± v (v > 0), or a couple of (complex conjugate) purely imaginary
eigenvalues ± iv (v > 0). In such a case, the assertions of Theorems 1 and 2 are expressed in terms of eigenvalues of matrices
A, B and parameters α, τ. In particular, this enables us to reformulate Theorems 1 and 2 in the case when the matrices A, B
are given in the corresponding Jordan canonical forms. Moreover, it seems that our results could be generalized, with some
technical adjustments, also for the matrix A having two distinct real eigenvalues. Under the above stated restrictions on
eigenvalues of the system matrices, a generalization into higher dimensions is possible as well.
6. Some applications
This section illustrates a simple use and applicability of Theorems 1 and 2. We start with
Example 1. We consider a fractional delay predator-prey system
Dαx1(t) = x1(t)[1 + x2(t) − x1(t − τ ) − x2(t)/u]
Dαx2(t) = x2(t)[−1 + x1(t) − x2(t − τ ) + x1(t)/u] (6.1)
where 0 < α, u < 1 and τ > 0 are real numbers, whose linearization (for linearization theorem on fractional differential equations,
we refer to [34]) along the unique positive equilibrium (u, u) leads to
Dαx1(t) = −ux1(t − τ ) + (u − 1)x2(t)
Dαx2(t) = (u + 1)x1(t) − ux2(t − τ ) (6.2)
(we note that this population model was studied in the first-order case α = 1 in [24,35]). To discuss its stability properties
with respect to the entries α, u and τ, we apply Theorem 2 with a = −u, b = u − 1 and c = u + 1. Thus, we get that the zero
solution to (6.2) is asymptotically stable if and only if
u
√
1 − u2
< cos(απ/2), (6.3)
or
u
√
1 − u2
≥
cos(απ/2)
cos(θ∗
1
)
and τ ∈ (0, τ∗
1,1), (6.4)
or
cos(απ/2)
cos(θ∗
m)
≤
u
√
1 − u2
<
cos(απ/2)
cos(θ∗
m−2
)
and τ ∈
(m−1)/2−1
j=−1
τ∗
2 j+1,−1, τ∗
2 j+3,1 (6.5)
where m ≥ 3 is an odd positive integer, τ∗
−1,−1
= 0 and τ∗
2 j+1,κ = τ∗
2 j+1,κ ( 1 − u2, −u) are given by (4.9) for j ≥ 0. The stability
boundary is depicted in Fig. 9 for α = 0.8 (appropriate stability region lies below this boundary) along with some significant
values of u. Some graphical comparisons of stability boundaries for various values of α are illustrated in Fig. 10. We can
observe here a delay-independent asymptotic stability case occurring when u < u∞ which corresponds to (6.3) (note that
this case cannot occur provided α = 1). For u ≥ u1 (i.e. when (6.4) occurs), we have a one-time stability loss with increasing
τ, while for u∞ ≤ u < u1 (when (6.5) occurs), a repeated stability switches appear. We point out that if u = u∞, then the zero
solution to (6.2) is asymptotically stable for all τ up to infinitely (countably) many values τ = τ∗
2 j+1,1
( j = 0, 1, . . . ) when it
is stable, but not asymptotically (note again that such a phenomenon cannot be obtained for the classical first-order model).
As it can be confirmed by Fig. 10, the delay-independent asymptotic stability is not actually present for α = 1, while the
conditions (6.4) and (6.5) agree with expectations based on [24, Corollary 3.2].
Finally, we choose the value u = 0.5 (i.e. u(1 − u2)−1/2 = 3−1/2), which was studied for α = 1 in [24], and discuss stability
properties of (6.2) with respect to changing τ and α. We can find three qualitatively different situations depending on α
with respect to two special values calculated numerically as α1 ≈ 0.6432790899 (in fact, cos(α1π/2) = 3−1/2 cos(θ∗
1
), see
(6.4) with respect to (4.4)) and α∞ ≈ 0.6081734480 (in fact, cos(α∞π/2) = 3−1/2, see (6.3)):
14 J. ˇCermák and T. Kisela / Commun Nonlinear Sci Numer Simulat 79 (2019) 104888
Fig. 9. The stability boundary for α = 0.8.
Fig. 10. The stability boundaries for various α.
• If α1 < α < 1, then the zero solution to (6.2) is asymptotically stable if and only if 0 < τ < τ∗
1,1
(it agrees with [24,
Remark 3.2] for α = 1);
• If α∞ ≤ α ≤ α1, then the zero solution to (6.2) is asymptotically stable if and only if τ ∈ (m−1)/2−1
j=−1
(τ∗
2 j+1,−1
, τ∗
2 j+3,1
)
where a positive odd integer m ≥ 3 depends on α and is uniquely determined by (6.5);
• If 0 < α < α∞, then the zero solution to (6.2) is asymptotically stable for all values of τ > 0.
To illustrate these results, we calculate the corresponding stability switches for particular values of α. The generic procedure
is as follows: For each specified α, we check if (6.3) or (6.4) holds (the latter one requires the knowledge of the root
θ∗
1
∈ (0, απ/2) to (4.4)). If none of them is valid, we sequentially calculate the roots θ∗
m ∈ (0, απ/2) to (4.4) for m = 3, 5, . . .
until we find m satisfying (6.5). Then we evaluate the corresponding τ∗
i,κ given by (4.9) for p = (0.75)1/2, q = −0.5, κ = ±1
and i = 1, 3, . . . , m. Thus we get:
• If α = 0.8, then the zero solution to (6.2) is asymptotically stable if and only if
τ ∈ (0, 1.781364151); (6.6)
• If α = 0.64, then m = 3 and the zero solution to (6.2) is asymptotically stable if and only if
τ ∈ (0, 3.589360507) ∪ (10.06793272, 10.77857157);
• If α = 0.615, then m = 5 and the zero solution to (6.2) is asymptotically stable if and only if
τ ∈ (0, 4.850664442) ∪ (7.976214463, 13.89430971) ∪ (21.45019407, 22.93795499).
J. ˇCermák and T. Kisela / Commun Nonlinear Sci Numer Simulat 79 (2019) 104888 15
Fig. 11. The phase trajectory of the solution to (6.1) with α = 0.8, u = 0.5, τ = 1.0 and initial conditions x1(t) ≡ 0.5 and x2(t) ≡ 0.6 on (−1, 0]; the stepsize
h = 0.03.
Fig. 12. The phase trajectory of the solution to (6.1) with α = 0.8, u = 0.5, τ = 3.0 and initial conditions x1(t) ≡ 0.5 and x2(t) ≡ 0.6 on (−3, 0]; the stepsize
h = 0.03.
A further decrease of α below the critical value α∞ ≈ 0.6081734480 already results into a delay-independent asymptotic
stability (i.e., the zero solution to (6.2) is asymptotically stable for all τ > 0).
Remark 4. We can supply the previous results also with respect to the original nonlinear problem (6.1). Figs. 11–14 depict
phase portraits of the solutions to (6.1) with α = 0.8, u = 0.5 and constant initial conditions x1(t) ≡ c1, x2(t) ≡ c2 (c1, c2 ∈ R)
prescribed on (−τ, 0] (for the used numerical algorithm based on the Adams method extended for fractional equations with
a delay see [36]).
In particular, Figs. 11 and 12 show how the choice of τ inside and outside the interval (6.6) results into stability and instability
of the positive equilibrium (0.5, 0.5), respectively. Further, Figs. 11, 13 and 14 illustrate the effect of initial conditions
gradually moving away from the equilibrium (0.5, 0.5) with respect to the asymptotic convergence of trajectories.
Now we show an application potential of Theorems 1 and 2 in stabilization problems utilizing a delayed feedback control.
Example 2. We consider the fractional differential system
Dαx1(t) = x2(t)
Dαx2(t) = ωx1(t) (6.7)
0 < α < 1 and ω = 0 being real numbers, whose characteristic equation λ2α − ω = 0 is identical with that for a basic differential
model
D2αx(t) − ωx(t) = 0
of the linear fractional oscillator. In the standard case ω < 0, the spectrum of the matrix system (6.7) is formed by a couple of
purely imaginary numbers μ1,2 = ±i
√
−ω lying inside the Matignon stability sector |arg(μ)| > π/2 for any 0 < α < 1. In the
16 J. ˇCermák and T. Kisela / Commun Nonlinear Sci Numer Simulat 79 (2019) 104888
Fig. 13. The phase trajectory of the solution to (6.1) with α = 0.8, u = 0.5, τ = 1.0 and initial conditions x1(t) ≡ 0.5 and x2(t) ≡ 2.0 on (−1, 0]; the stepsize
h = 0.03.
Fig. 14. The phase trajectory of the solution to (6.1) with α = 0.8, u = 0.5, τ = 1.0 and initial conditions x1(t) ≡ 0.5 and x2(t) ≡ 5.0 on (−1, 0]; the stepsize
h = 0.03.
inverted case ω > 0, a couple of real eigenvalues μ1,2 = ±
√
ω appears, hence the zero solution to (6.7) is not asymptotically
stable for any 0 < α < 1. Therefore, we investigate the stabilization of this zero equilibrium via appropriate diagonal delay
feedback control, i.e. we consider the system
Dαx1(t) = x2(t) + u1(t − τ )
Dαx2(t) = ωx1(t) + u2(t − τ )
under the control law ui(t) = kxi(t), i = 1, 2, where k is a real gain parameter and τ is a (positive) real time delay. Thus we
obtain the closed-loop system
Dαx1(t) = kx1(t − τ ) + x2(t)
Dαx2(t) = ωx1(t) + kx2(t − τ ) (6.8)
and wish to find conditions on stabilizing control parameters k, τ. Doing this, we apply Theorem 1 with a = k, b = 1 and
c = ω > 0 and get the conditions k +
√
ω < 0 and τ < τ∗(sgn(k)
√
ω, k) for asymptotic stability of the zero solution to (6.8).
In other words, the zero solution to the controlled system (6.8) is stabilized if and only if
k < −
√
ω and τ <
(2 − α)π/2 + arcsin(
√
ω/k sin(απ/2))
k2 − ω sin
2
(απ/2) +
√
ω cos(απ/2)
1/α
.
Thus, the stabilization is possible for all ω > 0 provided there are no additional restrictions on τ or k.
J. ˇCermák and T. Kisela / Commun Nonlinear Sci Numer Simulat 79 (2019) 104888 17
Fig. 15. The phase trajectory of the solution to (6.8) with α = 0.5, k = −3, ω = 1 and τ = 0.16; the stepsize h = 0.0016.
Fig. 16. The phase trajectory of the solution to (6.8) with α = 0.5, k = −3, ω = 1 and τ = 0.18; the stepsize h = 0.0018.
To illustrate our results, we present phase portraits of the solutions to (6.8) with α = 0.5, k = −3, ω = 1 (for the numerical
algorithm see again [36]). For this combination of parameters, the zero solution is stabilized when τ < 0.161. Thus,
Fig. 15 represents the stabilized case, while Fig. 16 shows the situation where the zero solution to (6.8) is not stabilized due
to an unsuitable choice of the control τ.
More generally, we can study this stabilization problem in a more complex form. Instead of (6.7), we consider a general
fractional differential system with an unstable equilibrium that we wish to stabilize via a diagonal delayed feedback control.
Thus we obtain the closed-loop system (2.1) where A is the Jacobi matrix of the uncontrolled system (evaluated at a
given equilibrium) and B = kI is the gain matrix (with a gain parameter k, I being the identity matrix). Stability analysis of
this controlled system leads to root analysis of quasi-polynomials of the type (3.7) where instead of the purely imaginary
coefficient we consider appropriate imaginary coefficient (with arbitrary real part). Such an analysis is a qualitatively more
difficult problem requiring some additional tools and can be viewed as an inspiration for the next research.
7. Concluding remarks
In this paper, the problem of delay-dependent stability switches for a linear planar autonomous fractional differential
system has been discussed. We have formulated stability criteria describing a nature of this phenomenon, including conditions
for its appearance as well as a number and exact calculations of corresponding stability switches. Contrary to the
conventional first-order case, a much more rich stability behaviour of a studied system has been observed, commented and
depicted. In particular, a delay-independent stability area has appeared for any derivative order between 0 and 1 (in the
first-order model, this area is empty). A border between delay-independent and delay-dependent stability areas has been
described explicitly in terms of entry parameters. In this specific case, the asymptotic stability property has been guaranteed
for almost all positive real time lags up to a uniform grid of isolated lags when non-asymptotic stability has occurred.
18 J. ˇCermák and T. Kisela / Commun Nonlinear Sci Numer Simulat 79 (2019) 104888
Also, we have explored dependence of stability areas on changing derivative order, especially with respect to the limit
values 1 and 0. Our analysis of these limit cases has shown that the derived stability criteria turn into the existing conditions
known for the appropriate first-order differential system and non-differential system, respectively. Besides these stability
investigations, an asymptotic result on algebraic decay rate of the solutions in the asymptotically stable case has been
presented.
As illustrated in the previous section, these results can be applied either in local stability analysis of nonlinear fractional
delay dynamical models, where the effect of stability switches appears, or in stabilization and synchronization problems of
control theory for fractional systems. In particular, a formulation of stabilizing conditions explicit with respect to a time
delay control provides a significant motivation for further research in this direction. The results presented in this paper
enable to formulate this type of stabilizing conditions for diagonal delayed feedback controls of unstable equilibria of some
fractional oscillators and other related models (where the eigenvalues of the appropriate Jacobi matrix are real or purely
imaginary numbers). The above indicated possible extension of such results to general fractional dynamical systems (with
arbitrary complex eigenvalues of the Jacobi matrix) could provide a very efficient tool for delayed feedback controls of
chaos (appearing, e.g., in the fractional Lorenz model of convective fluid) and other stabilization and synchronization issues
leading to the fractional delay system (2.1). We note that, under specific choices of the system matrices, (2.1) appears also
in PDα control of Newcastle robot (see [37], for outlines concerning more advanced models see also [38,39]). Our stability
procedures seem to be extendable also to this problem and thus may contribute to a more complex answer to stability
questions connected with this model.
Acknowledgements
The research has been supported by the grant 17-03224S of the Czech Science Foundation. The authors are grateful to
the anonymous referees for reading this paper and useful recommendations which helped to improve its content.
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Appendix G
Paper on higher-order two-term
FDDE [12] (CNSNS, 2023)
Previously in [13], we studied two-term FDDE of orders less than one. It is known
that the stability regions for first and second-order equations are qualitatively very
different, in this case a connected unbounded set with a boundary formed by a line
and a transcendental curve, in contrast to infinitely many touching triangles. Thus,
in [12] (co-author: J. Čermák; my author’s share 50 %), we aimed to describe the
transition between first-order and second-order two-term equations and perform a
comprehensive analysis of the expected stability switches.
Building on our previous experience and the proving techniques developed in
earlier works, we described the region of delay-independent stability, the stability
boundary, and provided a detailed analysis of stability switches. All of that in the
explicit form. We included all necessary details for precise calculations of their
number and values, which were then applied on the stabilization and destabilization
of fractional oscillators. The equation studied in this paper represents the simplest
case of real-valued problem where such a rich stability properties occur.
– 153 –
Communications in Nonlinear Science and Numerical Simulation 117 (2023) 106960
Contents lists available at ScienceDirect
Communications in Nonlinear Science and
Numerical Simulation
journal homepage: www.elsevier.com/locate/cnsns
Research paper
Stabilization and destabilization of fractional oscillators via a
delayed feedback control
Jan Čermák, Tomáš Kisela
∗
Institute of Mathematics, Brno University of Technology, Technická 2, CZ-616 69 Brno, Czech Republic
a r t i c l e i n f o
Article history:
Received 6 June 2022
Received in revised form 11 October 2022
Accepted 14 October 2022
Available online 20 October 2022
MSC:
34K37
34K20
93D15
Keywords:
Fractional oscillator
Fractional delay differential equation
Feedback control
Stabilization and destabilization
a b s t r a c t
This paper discusses the problem of stabilization and destabilization of fractional
oscillators by use of a delayed feedback control. A mathematical part of the problem
consists in stability analysis of appropriate fractional delay differential equations with
the derivative order varying between 1 and 2. Derived stability criteria are efficient and
easy to apply when stabilizing or destabilizing fractional oscillators in the standard as
well as inverted form. As a by-product of our results, we explicitly describe critical values
of a delay control parameter when stability property turns into instability and vice versa.
Evaluations of these stability switches are possible also in the limit harmonic case which
brings new insights into classical stability results on this topic.
© 2022 Elsevier B.V. All rights reserved.
1. Introduction
The model of a fractional oscillator originates from the equation of motion for the classical harmonic oscillator where
the second-order derivative (appearing in the acceleration term) is replaced by a fractional-order derivative between 1
and 2. Existing analysis of this fractional model revealed some similarities compared to a damped harmonic oscillator.
However, this damping property does not follow from friction (or other external sources) as in the classical harmonic
case, but from the internal structure of the fractional oscillator itself. For more details, including the cause of this intrinsic
damping, physical interpretations of fractional oscillators, and their response to some external forces, we refer to [1–4].
The decay of amplitudes of fractional oscillators is supported by another specific property, namely a weak oscillation.
While in the classical subcritically damped case, an oscillation around the equilibrium state occurs, the fractional oscillator
approaches the equilibrium from one side only (after a few overshoots) with oscillating velocity. Thus, from a certain
moment, its behavior begins to resemble rather critical or supercritical damping whose decay rate is not exponential, but
algebraic [5].
The classical damped harmonic oscillator has also its fractional extension. When studying the problem of the motion
of a rigid plate in a Newtonian fluid, the usefulness of a fractional damping term compared to classical damping was
justified theoretically as well as empirically in [6], and later analyzed in [7,8]. From the mathematical point of view, the
first derivative term (representing a damping proportional to velocity) was replaced by terms with rational derivative
orders (mostly 1/2 or 3/2). As it was shown in these and several following papers [9,10], this fractional model keeps
∗ Corresponding author.
E-mail addresses: cermak.j@fme.vutbr.cz (J. Čermák), kisela@fme.vutbr.cz (T. Kisela).
https://doi.org/10.1016/j.cnsns.2022.106960
1007-5704/© 2022 Elsevier B.V. All rights reserved.
J. Čermák and T. Kisela Communications in Nonlinear Science and Numerical Simulation 117 (2023) 106960
the same stability properties compared to the classical one, but has a different asymptotics. In particular, the fractional
damping term implies an algebraic decay of amplitudes to the equilibrium position.
Coming back to the fractional oscillator model, its fractional-order derivative guarantees asymptotic stability of the
system (as recalled above, this is not true for the oscillator with fractional damping). On the other hand, considering the
inverted oscillator (when the internal force linearly depending on the deflection is acting towards this deflection), the
asymptotic stability property does not hold for any derivative order between 1 and 2.
These observations can be extended via adding a suitable feedback control. The most simple feedback control is
that proportional to a current state of the system. However, it is easy to check that such a control cannot change
stability properties of neither harmonic nor fractional oscillator (including their inverted forms). Moreover, in the practical
implementation of feedback controls, it is very likely that time delays will occur. Therefore, it is important to understand
the sensitivity of the control system with respect to a time delay in the feedback loop. Depending on the values of a
delay, (asymptotic) stabilization or destabilization of many integer-order dynamical systems were reported. While general
reasons for a feedback stabilization consist in bringing the unstable (or even chaotic) system into a stable position, the
importance of a feedback destabilization appears in situations when we need to destabilize the stable, but in a certain
sense undesirable (e.g. pathological) state [11–13].
A mathematical platform for such feedback (de)stabilization of fractional models is provided by fractional delay
differential equations (FDDEs); for basics of the corresponding theory we refer, e.g. to [14–16]. Their stability analysis
currently belongs among rapidly developing research topics due to its practical as well as theoretical importance. However,
effective stability criteria for FDDEs are still rare (for some basic results we refer to [17–22]). From this viewpoint,
fractional and harmonic oscillators (whose position is at the border between stability and instability) represent very
suitable test models for stability analysis of their extended delayed feedback loops.
Following such outlines, we wish to discuss these problems: As a preliminary motivation, we consider the (undamped)
harmonic oscillator controlled via a time-delayed feedback loop that is linearly depending on the state of the system (but
not on its velocity). We wish to describe the structure of all delay and gain control parameters that enable either damping
of the oscillator (more precisely, its asymptotic stabilization around the equilibrium), or obversely its destabilization. As
the main problem, we consider the fractional oscillator (including its inverted form) with the same type of a control and
put similar questions on critical values of the control parameters, including a derivative order.
The paper is organized as follows. Section 2 introduces a short survey of some existing results, related notions and
methods. These methods involve, among others, techniques originating from root analysis of the appropriate characteristic
equation. A detailed description of locations of characteristic roots including their basic properties is provided in Section 3.
Section 4 presents stability criteria for studied linear FDDEs with the derivative order between 1 and 2. These results
indicate that when considering the derivative order as a varying bifurcation parameter, the first-order derivative can be
taken for the critical bifurcation value. More precisely, if the derivative order is changing between 0 and 1, the stability
areas display similar topological properties. However, exceeding the value 1, quite new features accompanying stability
investigations appear. Section 5 contains applications of these results to the problems of stabilization and destabilization of
fractional oscillators (including their inverted forms) via delayed feedback controls. As a by-product of these investigations,
we present appropriate results for the classical harmonic oscillator as well. The final section concludes the paper with a
survey of the presented results and possible perspectives.
2. A brief mathematical background
The dynamics of a fractional oscillator is given by the fractional differential equation
Dα
y(t) + ωα
y(t) = 0, t > 0 (2.1)
where α ∈ (1, 2) and ω > 0 is a parameter corresponding to frequency [3]. The symbol Dα
f (t) used here stands for the
Caputo fractional derivative which is, for any positive real α and a given function f , introduced as a composition of the
standard ⌈α⌉th order derivative (⌈·⌉ means a ceiling function) and the fractional (⌈α⌉ − α)th order integral
I⌈α⌉−α
f (t) =
∫ t
0
(t − ξ)⌈α⌉−α−1
Γ (⌈α⌉ − α)
f (ξ)dξ , t > 0,
i.e.
Dα
f (t) = I⌈α⌉−α d⌈α⌉
f (t)
dt⌈α⌉
, t > 0.
If we put I0
f (t) ≡ f (t) and α is a positive integer, then the Caputo derivative coincides with the standard (integer-order)
derivative (for more details on fractional operators we refer, e.g. to [23,24]). Thus, if particularly α = 2, then (2.1) becomes
the model for the classical harmonic oscillator
y′′
(t) + ω2
y(t) = 0, t > 0. (2.2)
It is well-known [2] that while (2.2) is non-asymptotically stable, its fractional analogue (2.1) is asymptotically stable
for any α ∈ (1, 2). In the inverted case, both the classical model
y′′
(t) − ω2
y(t) = 0, t > 0
2
J. Čermák and T. Kisela Communications in Nonlinear Science and Numerical Simulation 117 (2023) 106960
as well as its fractional analogue
Dα
y(t) − ωα
y(t) = 0, t > 0
are unstable.
To change stability or instability properties of these models, we introduce the control term u into their right-hand
sides. If we employ the basic feedback control u(t) = Ky(t), where K is a real gain parameter, then it is known (and
easy to verify) that its impact on stability properties of these models is limited. In particular, we are not able to achieve
asymptotical stabilization of (2.2) for any real K. We show that this situation changes if we consider the delay feedback
control of the form u(t) = Ky(t − τ) where, in addition to a gain parameter K, we employ also the real time lag τ > 0 as
the second control parameter.
From the mathematical point of view, we investigate stability properties of the FDDE
Dα
y(t) = ay(t) + by(t − τ), t > 0 (2.3)
where α ∈ (1, 2), τ > 0 and a, b are real parameters. Stability properties of (2.3) are known in both the limit integer-order
cases. It might be useful to recapitulate them [25,26].
Theorem 1. (i) Let α = 1. Then (2.3) is asymptotically stable if and only if either
a ≤ b < −a and τ is arbitrary,
or
|a| + b < 0 and τ <
arccos(−a/b)
(b2 − a2)1/2
.
(ii) Let α = 2 and b > 0. Then (2.3) is asymptotically stable if and only if a < 0 and there exists a non-negative even integer
ℓ such that
2ℓπ < τ
√
−a < (2ℓ + 1)π (2.4)
and
τ2
b < min(−(2ℓ)2
π2
− τ2
a, (2ℓ + 1)2
π2
+ τ2
a). (2.5)
Remark 1 (a). Both the parts (i) and (ii) were derived by use of root analysis of the corresponding characteristic
quasi-polynomials
P1(z) ≡ z − a − b exp(−zτ) and P2(z) ≡ z2
− a − b exp(−zτ), (2.6)
respectively. More precisely, the well-known stability condition says that (2.3) with α = 1 or α = 2 is asymptotically
stable if and only if all roots of (2.6)1 or (2.6)2, respectively, have negative real parts (see, e.g. [27]). A crucial problem,
namely how to convert this theoretical condition into an efficient form, was solved by use of the D-partition method [26]
(the case α = 1) and the Pontryagin criterion for location of quasi-polynomial roots in the open left complex
half-plane [25] (the case α = 2).
(b) The case b < 0 was not explicitly discussed in [25]. However, using similar arguments as those used for b > 0, the
extension of the part (ii) to b < 0 can be provided as well. We add that these results, even in a more efficient form, can
be also obtained as particular (limit) cases of our next considerations.
The above stated stability conditions for both the integer-order cases are of a quite different nature as illustrated by
Figs. 1 and 2 depicting appropriate stability regions in the (a, b)–plane.
Following both the integer–order cases, conditions for asymptotic stability of (2.3) originate from the requirement
on location of all characteristic roots left to the imaginary axis. More precisely, when considering (2.3), the generalized
characteristic quasi-polynomial takes the form
Pα(z) = zα
− a − b exp(−zτ)
and the corresponding theoretical stability condition can be stated as follows:
Lemma 1. Let α > 0, τ > 0 and a, b be real numbers.
(i) If all the roots of Pα(z) have negative real parts, then (2.3) is asymptotically stable.
(ii) If there exists a root of Pα(z) with a positive real part, then (2.3) is not stable.
The proof of this assertion originates from the Laplace transform method and can be found in several earlier papers
for α ∈ (0, 1) (see, e.g. [21, Lemma 1]). Its extension to arbitrary positive real values α uses an analogous argumentation.
Recently, results on effective stability conditions for (2.3) with α ∈ (0, 1) have appeared in [17,18,21]. The stability area
in the (a, b)-plane was described either via boundary parametric curves, or via implicit formulae involving the (unique)
3
J. Čermák and T. Kisela Communications in Nonlinear Science and Numerical Simulation 117 (2023) 106960
Fig. 1. Stability region of (2.3) for α = 1 and τ = 1.
Fig. 2. Stability region of (2.3) for α = 2 and τ = 1.
switch between stability and instability with respect to increasing τ (this type of description is used in Theorem 1 (i)
for α = 1). In both these descriptions, stability properties of (2.3) with α ∈ (0, 1) essentially share the same structure
as in the case α = 1. As Figs. 1 and 2 indicate, if α ∈ (1, 2), then the structure of stability properties of (2.3) become
qualitatively different. From this point of view, α = 1 can be viewed as an important bifurcation value with respect
to a changing derivative order. This fact brings another reason why to study (2.3): we wish to clarify a mathematical
background of this phenomenon and provide an interpolation between both the systems of stability conditions for (2.3)
(recalled in Theorem 1) with α continuously varying between 1 and 2.
3. Distribution of characteristic roots
In this section, we analyze location of characteristic roots of Pα(z) with respect to the imaginary axis (and discuss also
some related root properties). Thus we prepare a necessary mathematical apparatus for conversion of Lemma 1 into an
effective from.
In addition to some elementary and well-known root properties of Pα(z) (such as complex conjugacy), the series of
the following properties holds.
4
J. Čermák and T. Kisela Communications in Nonlinear Science and Numerical Simulation 117 (2023) 106960
Proposition 1. Let α ∈ (1, 2), τ > 0 and a, b be real numbers. Then z = r exp(iϕ) (r ≥ 0, ϕ ∈ (−π, π]) is a root of Pα(z)
with multiplicity greater than one if and only if either z = 0 or there exists an integer k such that αϕ − ϕ + τr sin(ϕ) = kπ
and τr sin(αϕ) + α sin(αϕ − ϕ) = 0. Moreover, any root of Pα(z) has multiplicity at most three.
Proof. Let z be a root of Pα(z) with multiplicity two. Then it also must hold
αzα−1
+ bτ exp(−zτ) = 0, (3.1)
and we get a = zα
+ α
τ
zα−1
, b = −α
τ
zα−1
exp(zτ). Since a, b are real numbers, the first part of the assertion follows by
elaborating these two expressions for zero imaginary parts with respect to z = r exp(iϕ) (r ≥ 0, ϕ ∈ (−π, π]).
Further, a triple root of Pα(z) satisfies, in addition to (3.1), also α(α−1)zα−2
−bτ2
exp(−zτ) = 0. Such a system admits
a unique solution z = (1 − α)/τ occurring if a = (1 − α)α−1
/τα
, b = −α(1 − α)α−1
exp(1 − α)/τα
(provided these values
are feasible). Similarly we can show that there are no roots of multiplicity greater than three. □
Proposition 2. Let α ∈ (1, 2), τ > 0, a < 0 and b be real numbers. Then there exists δ = δ(α, a) > 0 such that all the roots
z of Pα(z) satisfy | arg(z)| > π/2 whenever |b| < δ.
Proof. Let z = r exp(iϕ) where r ≥ 0 and |ϕ| < π/2. We substitute it into Pα(z) = 0 and rewrite the real and imaginary
parts as
rα
cos(αϕ) + |a| − b exp(−rτ cos(ϕ)) cos(rτ sin(ϕ)) = 0,
rα
sin(αϕ) + b exp(−rτ cos(ϕ)) sin(rτ sin(ϕ)) = 0,
which implies
(rα
− |a|)2
+ 2rα
|a|(1 + cos(αϕ)) = b2
exp(−2rτ cos(ϕ)). (3.2)
Since r ≥ 0 and |ϕ| < π/2, the left-hand side of (3.2) has a minimum equal either to a2
or a2
sin2
(αϕ) for |αϕ| ≤ π/2 or
|αϕ| > π/2, respectively. The right-hand side of (3.2) reaches its maximum for r = 0 and its value is b2
. Thus, (3.2) has
no solution for any b such that |b| < δ = |a| sin(απ/2) which implies the assertion. □
Lemma 2. Let α ∈ (1, 2), τ > 0 and a, b be real numbers. Then all functions z = z(a, b) defined implicitly by Pα(z) = 0 are
continuous whenever z(a, b) is not zero.
Proof. Denote z = r exp(iϕ) and let F and G be real and imaginary parts of Pα, respectively. For the sake of clarity, we
will use the notation Pα(a, b; z) to point out the dependence on parameters a, b. Then we can expand Pα(a, b; z) = 0 as
F(a, b; r, ϕ) ≡ rα
cos(αϕ) − a − b exp(−rτ cos(ϕ)) cos(rτ sin(ϕ)) = 0 ,
G(a, b; r, ϕ) ≡ rα
sin(αϕ) + b exp(−rτ cos(ϕ)) sin(rτ sin(ϕ)) = 0 .
The implicit function theorem states that if the fixed values ˆa, ˆb, ˆr, ˆϕ satisfy F(ˆa, ˆb; ˆr, ˆϕ) = 0, G(ˆa, ˆb; ˆr, ˆϕ) = 0 and
Fr (ˆa, ˆb; ˆr, ˆϕ)Gϕ(ˆa, ˆb; ˆr, ˆϕ)−Fϕ(ˆa, ˆb; ˆr, ˆϕ)Gr (ˆa, ˆb; ˆr, ˆϕ) ̸= 0, then there exists an open set O ∈ R2
containing (ˆa, ˆb) such that
there exist unique continuously differentiable functions r, ϕ such that r(ˆa, ˆb) = ˆr, ϕ(ˆa, ˆb) = ˆϕ and F(a, b; r(a, b), ϕ(a, b)) =
0, G(a, b; r(a, b), ϕ(a, b)) = 0 for all (a, b) ∈ O.
In our case, Fr Gϕ − FϕGr equals
r ·
[(
αrα−1
− τ|b| exp(−rτ cos(ϕ))
)2
+ 2αrα−1
τ|b| exp(−rτ cos(ϕ)) (1 + sgn(b) cos(rτ sin ϕ + αϕ − ϕ))
]
. (3.3)
Since z (then also r) is nonzero, (3.3) can be equal to zero provided the term in the square bracket is zero. Assume that
this does occur. Since both the terms in the square brackets are nonnegative, we obtain
|b| =
α
τ
rα−1
exp(rτ cos(ϕ)) and rτ sin(ϕ) + αϕ − ϕ =
(
2k +
1 + sgn(b)
2
)
π for a suitable k ∈ Z.
After some technical rearrangements, we can see that these conditions are identical to those for multiple roots described
in Proposition 1.
Thus we get that if ˆa, ˆb are fixed and ˆz is a nonzero root of Pα(ˆa, ˆb; z), i.e. ˆzα
− ˆa − ˆb exp(−ˆzτ) = 0, then there
exists a complex-valued function z(a, b) such that z(ˆa, ˆb) = ˆz and Pα(a, b; z(a, b)) = 0. By a continuous extension of the
corresponding open set O, we obtain that this function is unique and continuous for all points (a, b, z(a, b)) provided
z(a, b) is not a multiple root of Pα(a, b; z).
Pα(a, b; z) has countably many nonzero roots. It defines countably many branches of z(a, b) which do not intersect
each other unless a, b imply a multiple root z of Pα(a, b; z); in this case, the corresponding branches cross each other.
Since all the functions involved in Pα(a, b; z) are continuous at these points, we conclude that the branches z(a, b) do not
lose their continuity property there. □
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J. Čermák and T. Kisela Communications in Nonlinear Science and Numerical Simulation 117 (2023) 106960
Remark 2. Lemma 2 admits that the zero root occurring for a + b = 0 might not depend continuously on a, b. Indeed,
for values of α not admitting negative real roots of Pα(z), we can observe that the zero root ‘‘disappears’’ when crossing
the line a + b = 0.
Now we proceed to analysis of conditions ensuring that all the roots of Pα(z) have negative real parts. Doing it, we
employ the D-partition method. This method, when applied to Pα(z), originates from analytical descriptions of curves
in the (a, b)-plane such that Pα(z) admits either zero or purely imaginary roots just when its coefficients a, b are
lying on these curves. Such descriptions are independent of values of the parameter α and appeared in several earlier
papers [18,21]. It holds that Pα(z) has the zero root if and only if a + b = 0, and Pα(z) has a purely imaginary root
z = ±is/τ (s > 0) if and only if s ̸= jπ for any integer j and
a =
sα
sin(s + απ/2)
τα sin(s)
and b = −
sα
sin(απ/2)
τα sin(s)
. (3.4)
From the geometrical point of view, Pα(z) admits the zero or purely imaginary roots if and only if the couple (a, b) is lying
either on the line a + b = 0, or on some of the curves Γj (j = 0, 1 . . . ) given in the parametric forms
Γj : aj(s) =
sα
sin(s + απ/2)
τα sin(s)
,
bj(s) = −
sα
sin(απ/2)
τα sin(s)
,
s ∈ (jπ, jπ + π). (3.5)
This system of curves plays a crucial role in stability investigations of (2.3). Properties of these curves significantly differ
with respect to α ∈ (0, 1] or α ∈ (1, 2). In the sequel, we explore the latter case in a more detail.
Lemma 3. Let α ∈ (1, 2), τ > 0 be real numbers and let Γj (j = 0, 1 . . . ) be the curves defined by (3.5). Then it holds:
(i) The line a + b = 0 is tangent to the curve Γ0 at the origin, and the line
p−
0 : b = a −
(π
τ
)α
cos
(απ
2
)
is the asymptote to Γ0 as s → π−
. Moreover, b0(s) < 0, b0(s) < a0(s) − (π/τ)α
cos(απ/2) and b0(s) < −a0(s) for all
s ∈ (0, π).
(ii) If j is a positive odd integer, then Γj has asymptotes p+
j (as s → jπ+
) and p−
j (as s → (j + 1)π−
) given by
p+
j : b = a −
(
jπ
τ
)α
cos
(απ
2
)
and p−
j : b = −a +
(
jπ + π
τ
)α
cos
(απ
2
)
.
Moreover, bj(s) > 0, bj(s) > aj(s) − (jπ/τ)α
cos(απ/2) and bj(s) > −aj(s) + ((jπ + π)/τ)α
cos(απ/2) for all s ∈ (jπ, jπ + π).
(iii) If j is a positive even integer, then Γj has asymptotes p+
j (as s → jπ+
) and p−
j (as s → (j + 1)π−
) given by
p+
j : b = −a +
(
jπ
τ
)α
cos
(απ
2
)
and p−
j : b = a −
(
jπ + π
τ
)α
cos
(απ
2
)
.
Moreover, bj(s) < 0, bj(s) < −aj(s) + (jπ/τ)α
cos(απ/2) and bj(s) < aj(s) − ((jπ + π)/τ)α
cos(απ/2) for all s ∈ (jπ, jπ + π).
Proof. Verifications of all the three parts are of a technical nature. We derive the part (i), the remaining parts can be
proven analogously.
Let j = 0. Then lims→0+ (a0(s), b0(s)) = (0, 0) due to α > 1, and the form of the tangent line a + b = 0 easily follows
from the property lims→0+ b0(s)/a0(s) = −1. Similarly, the value of slope k and b-intercept q of the asymptote p−
0 to Γ0
(as s → π−
) follow from the standard formulae
k = lim
s→π−
b0(s)
a0(s)
= 1 and q = lim
s→π−
(b0(s) − a0(s)) = −
(π
τ
)α
cos
(απ
2
)
.
Further, the property b0(s) < 0 is evident. The second inequality appearing in the part (i) is equivalent to
sα
τα sin(s)
(
sin
(απ
2
)
+ sin(s) cos
(απ
2
)
+ cos(s) sin
(απ
2
))
>
(π
τ
)α
cos
(απ
2
)
,
and also to
sα
(
1 + tan
(απ
2
) 1 + cos(s)
sin(s)
)
< πα
due to s ∈ (0, π) and α ∈ (1, 2). The validity of the last inequality is easy to verify.
Similar calculations also imply b0(s) < −a0(s) for all s ∈ (0, π). □
6
J. Čermák and T. Kisela Communications in Nonlinear Science and Numerical Simulation 117 (2023) 106960
Fig. 3. The common asymptote to Γ1 and Γ2 and the corresponding trapezoids for α = 1.8 and τ = 1.
Remark 3. Lemma 3 implies that each of the curves Γj (j = 0, 1 . . . ) is contained in an infinite trapezoid bounded
by the a-axis and two straight lines (namely its asymptotes). These asymptotes intersect the a-axis at the values a =
((jπ + π)/τ)α
cos(απ/2). Also, we can see that each pair Γj, Γj+1 shares a common asymptote. The situation is depicted
in Fig. 3.
Lemma 4. Let α ∈ (1, 2), τ > 0 be real numbers, and let Γj (j = 0, 1 . . . ) be the curves defined by (3.5). Further, let
Xm,n = (am,n, bm,n) be intersections of Γm and Γn (if they exist). Then it holds:
(i) The intersection (am,n, bm,n) exists (and it is unique) if and only if m, n have the same parity.
(ii) am,m+2k < 0 for all k ∈ Z such that k > −m/2.
(iii) am,m+2k > am,m+2(k+1) for all k ∈ Z such that k > −m/2.
(iv) am,m+2k > am+2ℓ,m+2k+2ℓ for all k ∈ Z such that k > −m/2 and ℓ = 1, 2 . . . .
Proof. (i) By Lemma 3, the sign of functions bj(s) from (3.5) depends on parity of j. Hence, to find the intersections
between Γm and Γn, we put n = m + 2k for a suitable k ∈ Z and search for a couple (sm,m+2k, sm+2k,m) such that
sm,m+2k ∈ (mπ, mπ + π), sm+2k,m ∈ ((m + 2k)π, (m + 2k + 1)π) and
am(sm,m+2k) = am+2k(sm+2k,m), bm(sm,m+2k) = bm+2k(sm+2k,m).
Substituting (3.5) into these relations one gets
(sm,m+2k)α
sin(sm,m+2k + απ/2)
sin(sm,m+2k)
=
(sm+2k,m)α
sin(sm+2k,m + απ/2)
sin(sm+2k,m)
and
(sm,m+2k)α
sin(sm,m+2k)
=
(sm+2k,m)α
sin(sm+2k,m)
(3.6)
which leads to sin(sm,m+2k + απ/2) = sin(sm+2k,m + απ/2), hence
sm+2k,m = (2m + 2k + 3 − α)π − sm,m+2k.
Let sm,m+2k = mπ + ξ for a suitable ξ ∈ (0, π). Then using (3.6), intersections between Γm and Γm+2k can be expressed
via roots of the function
gm,m+2k(ξ) =
sin(απ + ξ)
sin(ξ)
−
(
(2m + 2k + 3 − α)π
mπ + ξ
− 1
)α
. (3.7)
Since limξ→0+ gm,m+2k(ξ) = ∞, limξ→π− gm,m+2k(ξ) = −∞ and g′
m,m+2k(ξ) < 0 for all ξ ∈ (0, π), the root ξm,m+2k of
gm,m+2k exists and it is determined uniquely.
(ii) Let sj be a root of aj(s), s ∈ (jπ, jπ + π). Obviously, this root is determined uniquely as sj = (j + 1 − α/2)π for all
j = 0, 1, . . . . Moreover, aj(s) > 0 for s < sj and aj(s) < 0 for s > sj. We put ξ0 = sj − jπ = (1 − α/2)π. The property (ii)
now follows from gm,m+2k(ξ0) > 0 for any m, k.
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J. Čermák and T. Kisela Communications in Nonlinear Science and Numerical Simulation 117 (2023) 106960
Fig. 4. Some intersections Xm,n = (am,n, bm,n) for α = 1.8, τ = 1 and m, n ∈ {0, 1, 2, 3, 4, 5, 6, 7}.
(iii) Since gm,m+2k(ξ) < gm,m+2(k+1)(ξ) for all ξ ∈ (0, π), the roots of these functions satisfy ξm,m+2k > ξm,m+2(k+1).
Moreover, aj(s) is monotonically decreasing whenever aj(s) < 0, which implies am,m+2k > am,m+2(k+1).
(iv) The statement is a consequence of the property (iii). Indeed, am,m+2k > am,m+2k+2ℓ = am+2k+2ℓ,m > am+2k+2ℓ,m+2ℓ =
am+2ℓ,m+2k+2ℓ. □
Remark 4. For the sake of lucidity, the location and ordering of intersections Xm,n = (am,n, bm,n) between Γm and Γn is
depicted in Fig. 4.
Lemma 5. Let α ∈ (1, 2), τ > 0, a < 0 and b be real numbers, and let a couple (a, b) ∈ Γj for a unique j. If |b| increases,
then a new root of Pα(z) with a positive real part appears.
Proof. Let z = z(a, b) be the corresponding purely imaginary root of Pα(z). By Lemma 2, there exists a neighborhood of
(a, b) such that z(a, b) is a continuous function on this neighborhood. Moreover, we can calculate
∂z(a, b)
∂b
=
exp(−zτ)
αzα−1 + bτ exp(−zτ)
=
zα
− a
b(αzα−1 + τzα − τa)
,
which is the well-defined expression because the root z = z(a, b) is simple. Substituting z = is (s > 0) and evaluating
the real part we get
R
(
∂z(a, b)
∂b
)
⏐
⏐
z=is
=
1
b
x1x3 + x2x4
x2
3 + x2
4
where x1 = sα
cos(απ/2) − a, x2 = sα
sin(απ/2), x3 = αsα−1
cos((α − 1)π/2) + τsα
cos(απ/2) − τa and x4 =
αsα−1
sin((α − 1)π/2) + τsα
sin(απ/2). If a < 0 then
x1x3 + x2x4 = τ(sα
+ a)2
− 2asα
τ(1 + cos(απ/2)) − αasα−1
sin(απ/2) > 0.
Since analogously we can dispose with the complex conjugate case z = −is (s > 0), unifying both the cases we get
sgn
(
∂R(z(a, b))
∂b
⏐
⏐
z=±is
)
= sgn(b)
which proves the assertion. □
4. Effective stability criteria for (2.3) with α ∈ (1, 2)
Lemma 1 supported by the results of Section 3 shows that the line a +b = 0 and the curves Γj given by (3.5) will play
a crucial role in the formulation of efficient stability conditions for (2.3). Before we state these conditions, some auxiliary
notation might be useful.
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J. Čermák and T. Kisela Communications in Nonlinear Science and Numerical Simulation 117 (2023) 106960
Fig. 5. Stability boundary Γ AS
and stability region Sα of (2.3) for α = 1.8 and τ = 1.
Let P be the line segment
a = −s, b = s, s ∈ (0, T)
where T = (3π − απ)α
/(2τα
| cos(απ/2)|), and let ˜Γj (j = 0, 1, . . . ) be the parts of Γj with the endpoints Xj,j−2 and Xj,j+2
given by its intersections with the neighboring curves Γj−2 and Γj+2 (see Fig. 4). If j = 0 or j = 1, then the curves Γ−2
and Γ−1 are not defined; in such a case, we introduce the corresponding endpoints as (0, 0) and (−T, T), respectively (for
analytical descriptions of all the other endpoints we refer to Lemma 4 and its proof). Further, we put
Γ AS
=
∞⋃
j=0
˜Γj ∪ P
and denote by Sα the open area in the (a, b)-plane containing the negative part of a-axis and bounded by Γ AS
. Also, we
set Uα = R2
\cl(Sα) where cl(·) means the closure of a given set. The area Sα (including some angular bounds of the curve
Γ AS
and a detail of the situation near the origin) is depicted in Figs. 5 and 6.
Then an effective reformulation of Lemma 1 is provided by
Theorem 2. Let α ∈ (1, 2), τ > 0 and a, b be real numbers.
(i) If (a, b) ∈ Sα, then (2.3) is asymptotically stable.
(ii) If (a, b) ∈ Uα, then (2.3) is not stable.
Proof. By Lemma 1, it is enough to show that if (a, b) ∈ Sα, then all the roots of Pα(z) have negative real parts, and if
(a, b) ∈ Uα, then there exists a root of Pα(z) with a positive real part.
Proposition 2 implies that there exists a neighborhood O of the negative a-axis where all the roots of Pα(z) have
negative real parts. In addition, all the roots of Pα(z) depend continuously on its parameters a, b due to Lemma 2. Hence,
the neighborhood O can be expanded until it is bounded by the appropriate parts of the line a + b = 0 and the curves
Γj (j = 0, 1 . . . ). The properties described in Lemmas 3 and 4 (supported by some simple calculations) yield that such an
expansion of O is bounded just by the curve Γ AS
. Finally, by Lemma 5, when crossing Γ AS
through any curve ˜Γj (while
expanding the neighborhood O), a root of Pα(z) with a positive real part appears. To complete the proof, it is enough to
employ an obvious property claiming that Pα(z) admits a root with the positive real part whenever a + b > 0. □
Remark 5. Geometry of the stability boundary Γ AS
can be described as follows: If a < 0 and b > 0, then this curve is
formed by a part of the line a + b = 0 connecting the origin and the curve Γ1. Then Γ AS
follows Γ1 until it is crossed by
Γ3, etc. If b < 0, then Γ AS
follows the curve Γ0 (that crosses the b-axis at b = −(π/τ − απ/(2τ))α
) until it is crossed by
Γ2, then follows appropriate part of Γ2 until reaching Γ4, etc.
Theorem 2 yields a geometric description of the stability area Sα of (2.3) in the (a, b)-plane. In the sequel, we provide
an alternative stability criterion for the case a < 0 that better agrees with the form of the conditions of Theorem 1 (the
9
J. Čermák and T. Kisela Communications in Nonlinear Science and Numerical Simulation 117 (2023) 106960
Fig. 6. Stability boundary Γ AS
and stability region Sα of (2.3) for α = 1.4 and τ = 1.
case a > 0 is not considered here because the corresponding stability conditions are quite straightforward as illustrated
in Figs. 5 and 6). On this account, for any α ∈ (1, 2), τ > 0, a < 0 and b being real numbers, we introduce the symbols
τ+
ℓ =
ℓπ + (2−α)π
2
+ arcsin
(⏐
⏐a
b
⏐
⏐ sin(απ
2
)
)
(
a cos(απ
2
) +
√
b2 − a2 sin2
(απ
2
)
)1/α
and τ−
ℓ =
ℓπ + π + (2−α)π
2
− arcsin
(⏐
⏐a
b
⏐
⏐ sin(απ
2
)
)
(
a cos(απ
2
) −
√
b2 − a2 sin2
(απ
2
)
)1/α
,
ℓ = 0, 1, 2 . . . . Then it holds
Theorem 3. Let α ∈ (1, 2), τ > 0, a < 0 and b be real numbers.
(i) If − sin(απ/2) < b/a < sin(απ/2), then (2.3) is asymptotically stable.
(ii) If b/a > sin(απ/2), then there exists an integer N1 ≥ 0 such that (2.3) is asymptotically stable for any τ ∈ (τ−
2k−2, τ+
2k),
and it is not stable for any τ ∈ (τ+
2k, τ−
2k+2) where k = 0, . . . , N1 (here we set τ−
−2 = 0, τ−
2N1+2 = ∞).
(iii) If −1 < b/a < − sin(απ/2), then there exists an integer N2 ≥ 0 such that (2.3) is asymptotically stable for any
τ ∈ (τ−
2k−1, τ+
2k+1), and it is not stable for any τ ∈ (τ+
2k+1, τ−
2k+3) where k = 0, . . . , N2 (here we set τ−
−1 = 0, τ−
2N2+3 = ∞).
(iv) If b/a < −1, then (2.3) is not stable.
Proof. By (3.4),
b
a
= −
sin(απ/2)
sin(s + απ/2)
. (4.1)
This immediately implies the property (i) because all the couples (a, b) lying on Γ AS
have to satisfy |b/a| ≥ sin(απ/2).
The proofs of the parts (ii), (iii) are technical. The formulae for τ+
ℓ , τ−
ℓ can be derived via expressing s from (4.1) in the
form
s = (−1)ℓ+1
arcsin
(a
b
sin
(απ
2
))
+ ℓπ −
απ
2
,
using its proper sign analysis, substitution into (3.4) and some straightforward rearrangements. The existence of the values
N1, N2 follows from the property of the intersections Xj,j+2 = (aj,j+2, bj,j+2) between the curves Γj, Γj+2 discussed in
Lemma 4. Indeed, the ratio
⏐
⏐
⏐
⏐
bj,j+2
aj,j+2
⏐
⏐
⏐
⏐ =
sin(απ/2)
| sin(sj,j+2 + απ/2)|
is monotonically decreasing and tending to sin(απ/2) as j → ∞ because the sequence {sj,j+2 − jπ}∞
j=0 is decreasing and
its limit equals (3 − α)π/2 due to (3.7) (see the proof of Lemma 4).
The property (iv) is a direct consequence of the fact that the line a + b = 0 forms a part of Γ AS
. □
10
J. Čermák and T. Kisela Communications in Nonlinear Science and Numerical Simulation 117 (2023) 106960
Remark 6. It might be interesting to discuss the conclusions of Theorem 2 with α → 2−
and compare them with the
stability conditions of Theorem 1 (ii) derived for α = 2. To emphasize dependence on the derivative order α, we denote
the curves from (3.5) as Γα,j and their asymptotes (described in Lemma 3) as p+
α,j, p−
α,j throughout this remark.
We describe the calculation with j being odd. First, we determine the distance between the curve Γα,j and its
asymptotes p+
α,j, p−
α,j in the sense of the supremum metric. On this account, we introduce the symbol dI (Γα,j, p) as
the difference between b–coordinates of Γα,j and a line p maximized over all a ∈ I. Further, we denote a+
j =
(jπ)α
/τα
cos(απ/2), a−
j = (jπ +π)α
/τα
cos(απ/2), and let s+
j ∈ (jπ, jπ +π) and s−
j ∈ (jπ, jπ +π) be such that a(s+
j ) = a+
j
and a(s−
j ) = a−
j , respectively. Using this notation, we wish to show that d[a+
j
,∞)(Γα,j, p+
α,j) → 0 as α → 2−
. Indeed, it holds
d[a+
j
,∞)(Γα,j, p+
α,j) = sup
t∈(jπ,s+
j
]
sα
τα sin(s)
(
sin
(
t +
απ
2
)
+ sin
(απ
2
))
−
(
jπ
τ
)α
cos
(απ
2
)
= sup
t∈(jπ,s+
j
]
sα
− (jπ)α
τα
⏐
⏐ cos
(απ
2
) ⏐
⏐ −
sα
τα
sin
(απ
2
) 1 + cos(s)
sin(s)
=
(s+
j )α
− (jπ)α
τα
⏐
⏐ cos
(απ
2
) ⏐
⏐ −
(s+
j )α
τα
sin
(απ
2
) 1 + cos(s+
j )
sin(s+
j )
due to the monotony property of the maximized expression on the interval (jπ, s+
j ]. Moreover, by use of the identity
((s+
j )α
− (jπ)α
)| cos(απ/2)|/ cos(s+
j ) = (s+
j )α
sin(απ/2)/ sin(s+
j ), we obtain
d[a+
j
,∞)(Γα,j, p+
α,j) =
((s+
j )α
− (jπ)α
)
⏐
⏐ cos
(απ
2
) ⏐
⏐
τα| cos(s+
j )|
.
Since s+
j → jπ as α → 2−
, we actually get d[a+
j
,∞)(Γα,j, p+
α,j) → 0 as α → 2−
. Similarly, we can show that
d(−∞,a−
j
](Γα,j, p−
α,j) → 0 as α → 2−
and, if we denote by p∗ the a-axis, then also d[a+
j
,a−
j
](Γα,j, p∗) → 0 as α → 2−
. A
similar line of arguments leads to analogous results also for j being even.
Note that the asymptotes p+
α,j, p−
α,j themselves tend to the lines b = −a − (jπ)2
/τ2
and b = a + (jπ)2
/τ2
as α → 2−
.
Hence, one can observe that the stability area described in Theorem 2 actually approaches (when α → 2−
) the system
of triangles bounded by these lines and the negative a-axis (see Fig. 2). Thus the stability conditions of Theorem 2 are
converting to those of Theorem 1 (ii) when α → 2−
.
This fact provides another interesting consequence. If we make this limit transition also in Theorem 3 (serving as
an alternative description of Sα using τ as a driving parameter), then we can easily deduce that (2.3) with α = 2 is
asymptotically stable if and only if 0 < |b| < −a and
ℓπ
√
−a − |b|
< τ <
(ℓ + 1)π
√
−a + |b|
(4.2)
where ℓ is a non-negative integer that is even for b > 0 and odd for b < 0. This form of conditions seems to be more
effective compared to that of Theorem 1 (ii), especially with respect to explicit evaluations of stability switches for a
varying delay parameter.
5. Stabilization and destabilization of harmonic and fractional oscillator
Now we apply conclusions of the previous section to the problem stated in the introductory part. First, we consider
the classical harmonic oscillator and show how it can be asymptotically stabilized or destabilized via a feedback control
depending only on a position of the controlled object. Although these results can be partially derived from Theorem 1 (ii),
we deduce them more straightforwardly from (4.2). Then we study the linear fractional oscillator (which is, in the
uncontrolled case, asymptotically stable for any derivative order between 1 and 2) and describe conditions for its
destabilization. Also, we consider both these types of oscillators in their inverted forms and discuss conditions for their
stabilizations.
5.1. Delayed feedback control of a harmonic oscillator
We consider a controlled harmonic oscillator in the form
y′′
(t) + ω2
y(t) = u(t), t > 0 , (5.1)
ω > 0 is a real number representing frequency. Based on an elementary analysis, the uncontrolled linear oscillator (when
u is identically zero) is non-asymptotically stable and cannot be asymptotically stabilized via a feedback control of the
11
J. Čermák and T. Kisela Communications in Nonlinear Science and Numerical Simulation 117 (2023) 106960
Fig. 7. The stability region for harmonic oscillator (5.1) with control (5.2) depicted in the (ω2
, K)-plane for τ = 1.
form u(t) = Ky(t) for any real gain parameter K (obviously, it can be destabilized whenever K ≥ ω2
). When implementing
a time lag into the controller, i.e. when
u(t) = Ky(t − τ), (5.2)
the situation becomes different. There exists a nonempty set of couples (K, τ) such that (5.1)–(5.2) is either asymptotically
stable or unstable. To describe these sets, we can set a = −ω2
and b = K in (2.4), (2.5) and obtain their analytical
descriptions explicitly depending on the gain parameter K. More precisely, (5.1)–(5.2) is asymptotically stable if and only
if there exists a non-negative integer ℓ such that
2ℓπ/τ < ω < (2ℓ + 1)π/τ and |K| < min
(
ω2
− (ℓπ/τ)2
, ((ℓ + 1)π/τ)2
− ω2
)
where ℓ is even or odd for K positive or negative, respectively (see Fig. 7 for the corresponding stabilization region).
If we need to evaluate an explicit dependence of stability of (5.1)–(5.2) on a time lag parameter, then (4.2) provides a
more suitable platform. Indeed, (4.2) with a = −ω2
and b = K immediately implies that if the gain K is fixed such that
0 < |K| < ω2
, then (5.1)–(5.2) is asymptotically stable if and only if
ℓπ
√
ω2 − |K|
< τ <
(ℓ + 1)π
√
ω2 + |K|
(5.3)
where ℓ is a non-negative integer that is even for K > 0 and odd for K < 0. In addition, we can notice that the inequality
ℓπ
√
ω2 − |K|
<
(ℓ + 1)π
√
ω2 + |K|
can actually occur provided ℓ is less than the value
ℓ∗
=
ω2
− |K| +
√
ω4 − K2
2|K|
that enables to determine the number of such stability switches.
Quite analogously, as a counterpart of the condition (5.3), we can formulate the conditions for destabilization of the
harmonic oscillator via the control (5.2).
Finally, it may be useful to describe this asymptotic stabilization or destabilization area in the domain (K, τ) of both
the control parameters. Appropriate boundary curves τℓ = τℓ(K) can be obtained directly from (5.3) in the form
τℓ =
ℓπ
√
ω2 + (−1)ℓ+1K
, −ω2
< K < ω2
, ℓ = 0, 1, . . .
and they are depicted in Fig. 8.
5.2. Delayed feedback control of an inverted linear oscillator
We consider the controlled inverted linear oscillator
y′′
(t) − ω2
y(t) = u(t), t > 0 (5.4)
that is for u = 0 unstable. Obviously, using the control u(t) = Ky(t), it can be stabilized (only non-asymptotically) provided
K < −ω2
(in fact, thus we convert (5.4) into the standard harmonic case). If we employ the control (5.2), then our previous
analysis easily shows that asymptotic stabilization of (5.4) is not possible for any couple (K, τ).
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J. Čermák and T. Kisela Communications in Nonlinear Science and Numerical Simulation 117 (2023) 106960
Fig. 8. The stability region for harmonic oscillator (5.1) with control (5.2) depicted in the (K, τ)-plane for ω = 1.
5.3. Delayed feedback control of a fractional oscillator
Now we consider the controlled fractional oscillator
Dα
y(t) + ωα
y(t) = u(t), t > 0 (5.5)
where α ∈ (1, 2) and ω > 0 are real numbers. It is well-known [2] that this fractional model considered without a control
term is asymptotically stable for any α ∈ (1, 2). Thus, using our results from the previous section, we can discuss its
possible destabilization via (5.2).
By Theorem 3, if |K| < ωα
sin(απ/2), then (5.5) remains asymptotically stable for any non-negative value of τ, hence
its destabilization is possible only when |K| ≥ ωα
sin(απ/2). If, particularly, K > ωα
, then (5.5) is destabilized via (5.2)
unconditionally, i.e. for any τ > 0. For other values of K, the stability behavior depends on τ. If ωα
sin(απ/2) < K < ωα
,
then (5.5) becomes destabilized via (5.2) provided
K >
sα
ω,ℓ sin(απ/2)
τα sin(sω,ℓ)
for a non-negative odd integer ℓ, (5.6)
sω,ℓ being determined uniquely by ωα
= −sα
ω,ℓ sin(sω,ℓ + απ/2)/(τα
sin(sω,ℓ)), sω,ℓ ∈ (ℓπ, ℓπ + π). If K < −ωα
sin(απ/2),
then (5.5) becomes destabilized via (5.2) provided
K <
−sα
ω,ℓ sin(απ/2)
τα sin(sω,ℓ)
for a non-negative even integer ℓ, (5.7)
sω,ℓ being introduced as above. The corresponding stability region in the (ωα
, K)-plane is depicted in Fig. 9.
The use of criteria (5.6) and (5.7) requires a numerical solving of nonlinear equations which negatively affects their
practical usability. Therefore, we describe the destabilization boundary in the (K, τ)–plane since it can provide explicit
formulae. Indeed, using Theorem 3 (parts (ii) and (iii)), we can evaluate stability switches with respect to a time lag. First,
we determine the number ℓ∗
of stability switches via roots ξj,j+2 of gj,j+2(ξ) = 0 (see (3.7)) for j odd (if K > 0) and even
(if K < 0). The value ℓ∗
is given uniquely by
| sin(ξℓ∗,ℓ∗+2 + απ/2)| ≥
ωα
|K|
sin(απ/2) and | sin(ξℓ∗−2,ℓ∗ + απ/2)| <
ωα
|K|
sin(απ/2).
Then (5.5) is destabilized via (5.2) when
τ+
ℓ < τ < τ−
ℓ , ℓ < ℓ∗
,
τ > τ+
ℓ∗
where ℓ is odd (if K > 0) or even (K < 0), and τ+
ℓ , τ−
ℓ are defined by Theorem 3. Similarly as in the harmonic case α = 2,
formulae for τ+
ℓ , τ−
ℓ can be interpreted as parts of the stability boundary curve τ = τ(K) in the (K, τ)-plane (see Fig. 10).
5.4. Delayed feedback control of an inverted fractional oscillator
Finally, we consider the controlled inverted fractional oscillator
Dα
y(t) − ωα
y(t) = u(t), t > 0 (5.8)
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J. Čermák and T. Kisela Communications in Nonlinear Science and Numerical Simulation 117 (2023) 106960
Fig. 9. The stability region for fractional oscillator (5.5) with control (5.2) depicted in the (ωα
, K)-plane for the values α = 1.8 and τ = 1.
Fig. 10. The stability region for fractional oscillator (5.5) with control (5.2) depicted in the (K, τ)-plane for the values α = 1.8 and ω = 1.
that is unstable if u = 0. Introducing the control term (5.2) one can stabilize this system as demonstrated by the stability
region in Fig. 11 (note that stabilization is not possible for α = 2). Unlike the classical fractional oscillator case, there are
no stability switches. The boundary curve for the asymptotic stability region in the (K, τ)-plane can be described as
τ(K) =
(2−α)π
2
+ arcsin
(ωα
K
sin(απ
2
)
)
(
ωα cos(απ
2
) +
√
K2 − ω2α sin2
(απ
2
)
)1/α
, K ≤ −ωα
.
Asymptotic stabilization of (5.8) via (5.2) then occurs whenever τ < τ(K) as depicted in Fig. 12.
6. Concluding remarks
We have discussed the problems connected with (asymptotic) stabilization and destabilization of fractional oscillators
via a delayed feedback loop. The mathematical background of the problem consisted in analysis of appropriate FDDEs.
Stability properties of these equations with the derivative order between 0 and 1 were described previously. Our analysis
showed that the topological structure of stability regions (considered in the parameter space) significantly changes when
the derivative order exceeds the value 1. In particular, a repeated occurrence of finitely many stability switches with
respect to a changing delay parameter is a product of this analysis.
The derived stability criteria are easy to apply and result into effective conditions on control parameters ensuring
required asymptotic stabilization or destabilization of the studied models. In the limit case, when the fractional oscillator
becomes the harmonic one, stability switches can be evaluated fully explicitly. Thus, our stability criteria offer a new
computational view on the existing results also in this classical case.
14
J. Čermák and T. Kisela Communications in Nonlinear Science and Numerical Simulation 117 (2023) 106960
Fig. 11. The stability region for inverted fractional oscillator (5.8) with control (5.2) depicted in the (ωα
, K)-plane for the values α = 1.8 and τ = 1.
Fig. 12. The stability region for inverted fractional oscillator (5.8) with control (5.2) depicted in the (K, τ)-plane for the values α = 1.8 and ω = 1.
This research can be continued towards more advanced types of (nonlinear) fractional oscillators and controls of
their stability or oscillatory properties. It seems to be unavoidable that the development of techniques for analysis of
appropriate (nonlinear) FDDEs will require new approaches.
CRediT authorship contribution statement
Jan Čermák: Conceptualization, Methodology, Calculations, Investigation, Validation, Writing – review & editing.
Tomáš Kisela: Conceptualization, Visualisation, Methodology, Calculations, Investigation, Writing – review & editing.
Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have
appeared to influence the work reported in this paper.
Data availability
No data was used for the research described in the article.
Acknowledgments
The research has been supported by the grant GA20-11846S of the Czech Science Foundation.
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J. Čermák and T. Kisela Communications in Nonlinear Science and Numerical Simulation 117 (2023) 106960
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