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Continuation of homoclinic orbits Previous: Mathematical definition   Contents
During HSN continuation, only one bifurcation is tested for, namely the non-central
homoclinic-to-saddle-node orbit or NCH. This orbit forms the transition between HHS
and HSN curves. The strategy
used for detection is taken from HomCont [23].
During HHS continuation, all bifurcations detected in HomCont are also detected
in our implementation. For this, mostly test functions from [23] are used.
Suppose that the eigenvalues of fx(x0,a0) can be ordered according to
Â(mns) £ ... £ Â(m1) < 0 < Â(l1) £ ... £ Â(lnu), |
| (97) |
where Â() stands for 'real part of', ns is the number of stable, and nu the number of unstable eigenvalues.
The test functions for the bifurcations are
- Neutral saddle, saddle-focus or bi-focus
If both m1 and l1 are real, then it is a neutral saddle, if one is real and one consists of a pair of complex conjugates, the bifurcation is a saddle-focus, and it is a bi-focus when both eigenvalues consist of a pair of complex conjugates.
- Double real stable leading eigenvalue
- Double real unstable leading eigenvalue
- Neutrally-divergent saddle-focus (stable)
y = Â(m1) + Â(m2) + Â(l1) |
|
- Neutrally-divergent saddle-focus (unstable)
y = Â(m1) + Â(l2) + Â(l1) |
|
- Three leading eigenvalues (stable)
- Three leading eigenvalues (unstable)
- Non-central homoclinic-to-saddle-node
- Shil'nikov-Hopf
- Bogdanov-Takens point
For orbit- and inclination-flip bifurcations, we assume the same ordering of the eigenvalues of fx(x0,a0) = A(x0,a0) as shown in (97), but also that the leading eigenvalues m1 and l1 are unique and real:
Â(mns) £ ... £ Â(m2) < m1 < 0 < l1 < Â(l2) £ ... £ Â(lnu) . |
|
Then it is possible to choose normalised eigenvectors p1s and p1u of AT(x0,a0) and q1s and q1u of A(x0,a0) depending smoothly on (x0,a0), which satisfy
The test functions for the orbit-flip bifurcations are then:
- Orbit-flip with respect to the stable manifold
y = e-m1 T < p1s, x(1) - x0 > |
|
- Orbit-flip with respect to the unstable manifold
y = el1 T < p1u, x(0) - x0 > |
|
For the inclination-flip bifurcations, in [23] the following test functions are introduced:
- Inclination-flip with respect to the stable manifold
- Inclination-flip with respect to the unstable manifold
where f (f Î C1([0,1],\mathbb Rn)) is the solution to the adjoint system, which can be written as
|
ì ï ï ï í
ï ï ï î
|
|
×
f
|
(t) + 2 T AT(x(t),a0) f(t) = 0 |
|
|
|
|
ó õ
|
1
0
|
|
~
f
|
T
|
(t)[f(t)- |
~
f
|
(t)]dt = 0 |
|
|
|
| (98) |
where Ls and Lu are matrices whose columns form bases for the stable and unstable eigenspaces of A(x0,a0), respectively, and the last condition selects one solution out of the family cf(t) for c Î \mathbb R. Lu is equivalent to QU from the mathematical definition of the system, and Ls to QS. In the homoclinic defining system the orthogonal complements of QS and QU are used; in the adjoint system for the inclination-flip bifurcation, we use the matrices themselves (or at least, their transposed versions).
12.4.2 MatCont: the Koper example
Consider the following system of differential equations:
This system, which is a three-dimensional van der Pol-Duffing oscillator, has been introduced
and studied by [23]. It is used as a standard demo in HomCont.
Parameters e1 and e2 are kept at
0.1 and 1, respectively. We note that system (99) has certain symmetry:
If (x(t),y(t),z(t)) is a solution for a given value of l, then (-x(t),-y(t),-z(t))
is a solution for -l.
Starting from a general point (0, -1, 0.1) and setting k = 0.15 and l = 0,
we find by time integration a stable equilibrium at (-1.775, -1.775, -1.775).
By equilibrium continuation with l free, we find two limit points ( LP), at
(-1.024695, -1.024695, -1.024695) for l = -2.151860 with normal form coefficient
a=-4.437056e+000 and at
(1.024695, 1.024695, 1.024695) for l = 2.151860 with normal form coefficient
a=-4.437060e+000 (note the
reflection).
By continuation of the limit points with (k,l) free, MatCont detects a
cusp point CP at (0,0,0) for k = -3 and l = 0.
Also detected are two
Zero-Hopf points ZH for k = -0.3 at ±(0.948683, 0.948683, 0.948683)
and l = ±1.707630, but these are in fact Neutral Saddles. Further, two Bogdanov-Takens points BT are found for k = -0.05 at
±(0.991632, 0.991632, 0.991632) and l = ±1.950209. The normal form
coefficients are (a,b)=(6.870226e+000, 3.572517e+001).
A bifurcation diagram of (99) is shown in Figure .
Figure 32: Left: equilibrium bifurcation diagram of the Koper system.
Dashed line = equilibria, full line = limit points, CP = Cusp Point, BT =
Bogdanov-Takens, ZH = Zero-Hopf, LP = Limit Point.
Right: zoom on the left part of the diagram.
We now compute a HHS curve starting at one of the BT points, and setting
the curve-type to Homoclinic. To further initialize the homoclinic continuation,
some parameters have to be set and/or selected in the Starter window.
First we select 2 system parameters as free parameters, e.g. k and l.
We must also set the Initial amplitude, i.e. an approxinate size of the first homoclinic orbit.
The default setting of 1e-2 is a good value in most cases.
Another option is to start from a limit cycle with large period.
For example, select the BT point at l = 1.950209, and then compute
a curve of Hopf points H passing through it, along which one encounters a Generalized
Hopf bifurcation GH.
Stop the continuation at (the random value) l = 1.7720581, where the Hopf point
is at (0.98526071, 0.98526071, 0.98526071) and k = -0.23069361.
We then can find limit cycles for very slowly decreasing values of l,
(l decreases down to 1.77178), with a rapid increase in the period.
At some point, one can stop the continuation, and switch to the continuation of the
homoclinic orbit.
Figure 33: Left: limit cycles, starting from a Hopf point H and approaching
a homoclinic orbit. Right: a family of HHS orbits.
One can also monitor the eigenvalues of the equilibrium during continuation, by displaying
them in the Numerical window. This is a very useful feature, because it gives indications on what
further bifurcations might be expected.
For example, a non-central homoclinic-to-saddle-node reveals itself by the fact that
one eigenvalue approaches zero. Once one is close enough to such a point, the user can
switch to the continuation of HSN
orbits from there.