Course Control and System Theory of Rational Systems Motivated by
the Life Sciences
Homeworkset 4
Date issued: 4 October 2018.
Date due: 11 October 2018.
1. Polynomials with positive coefficients. Consider the system you have selected for Homework
Set 1. For polynomial biochemical reaction systems, the appropriate set of polynomials is
R+[X1, . . . , Xn]. The properties of the positive real numbers differ from those of the real
numbers R.
Can one add two polynomials of R+[X1, . . . , Xn] and obtain a product in the same set?
Can one multiply two such polynomials in the set? What are the additive and multiplicative
identity polynomials? For any polynomial p ∈ R+[X1, . . . , Xn] does there exist an additive
inverse in the set, thus a polynomial q ∈ R+[X1, . . . , Xn] such that p(x) + q(x) = 0? Does
there exist a multiplicative inverse in the set R+[X1, . . . , Xn]? With which real numbers can
one multiply a polynomial in R+[X1, . . . , Xn] and remain in the same set?
2. Rational functions. Consider the set of rational functions denoted by R[X1, . . . , Xn]. Which
conditions are needed for a rational function in this set to have a multiplicative inverse?
3. Differential equations of positive rational systems. Consider the ordinary differential equation
of rational form,
dx(t)
dt
=
c1x(t)
1 + c2x(t)
= f(x(t)), x(0) = 1,
c1, c2 ∈ R, c1 < 0, c2 > 0, X = R+.
Is the rational function of the differential equation nonsingular? What are conditions for the
existence and for the uniqueness of a solution of this differential equation? Is the function f
bounded over the set R+? Calculate the limit limx→∞
c1x
1+c2x. Does the limit limt→∞ x(t; 0, x0)
exist and, if so, what value takes the limit?
4. Rational differential equation. Consider the rational differential equation,
dx(t)
dt
=
c3x(t)
(x(t) − 1)(x(t) − 4)
, x(0) = x0 ∈ R.
Discuss the solution of this differential equation for various parts of the real numbers R.
Write or display the phase portrait of this system.
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Reading advice for Lecture 4
Please read of the lecture notes of Chapter 5 the Sections 5.3, 5.4, 5.5, and 5.6. Note that part of
these sections are not yet written.
Reading advice for the future Lecture 5
Lecture 5 will be presented on Tuesday 9 October. Please read of the lecture notes the Sections 6.1,
6.2, and 6.3.
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