# EE6415: Nonlinear Systems Analysis Tutorial 3 solution

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1. Find and classify the bifurcation for the system:
x˙ = y − 2x
y˙ = µ + x
2 − y

2. Classify the bifurcation for the system below, and draw its bifurcation diagram:
x˙ = µx + y + sin(x)
y˙ = µ + x − y

3. Classify the bifurcations in each of the following systems as µ varies
(a) x˙ 1 = x2; ˙x2 = µ(x1 + x2) − x2 − x
3
1 − 3x
2
1×2
(b) x˙ 1 = x2; ˙x2 = µ − x2 − x
2
1 − 2x1x2
(c) x˙ 1 = x2; ˙x2 = µ(x1 + x2) − x2 − x
3
1 + 3x
2
1×2

4. The following system is a model for a genetic control system.
x˙ = −ax + y
y˙ =
x
2
1 + x
2
− by
where a, b > 0. Analyze the bifurcations that occur as the parameter ”a” is varied and find the
critical value of ”a” in terms of the parameter b.

5. Consider the system
x˙ 1 = ax1 − x1x2
x˙ 2 = bx2
1 − cx2
where a, b, c are positive constants, and c > a. Let D = {x ∈ R
2
|x2 ≥ 0}.
(a) Show that every trajectory starting in D stays in D for all future time.
(b) Show that there are no periodic orbits in D.

6. For each of the following systems, show that the system has no limit cycles
(a) x˙ 1 = −x1 + x2; ˙x2 = g(x1) + ax2 a ̸= 1
(b) x˙ 1 = −x1 + x
3
1 + x1x
2
2
; ˙x2 = −x2 + x
3
2 + x
2
1×2
(c) x˙ 1 = 1 − x1x
2
2
; ˙x2 = x1

7. A model that is used to analyze chemical oscillators is given by
x˙ 1 = a − x1 −
4x1x2
1 + x
2
1
, x˙ 2 = bx1

1 −
x2
1 + x
2
1

where x1, x2 are dimensionless concentrations of certain chemicals, and a, b are positive constants.
(a) Prove that the system has a periodic orbit, when b < 3a/5 − 25/a
(b) Find and classify the bifurcations that occur as b varies, with a fixed a.

8. Consider the system
x˙ 1 = x2
x˙ 2 = −(2b − g(x1))ax2 − a
2×1
where a, b are positive constants, and
g(x) = (
0, |x| > 1
k, |x| ≤ 1
(a) Show that there are no periodic orbits for k < 2b.
(b) Show that there exists a periodic orbit for k > 2b.

9. Using a programming language of your choice(Matlab would be the easiest), create a video showing
the evolution of the phase portrait with µ for any system of your choice that displays
(b) Transcritical Bifurcation
(c) Supercritical Pitchfork Bifurcation
(d) Subcritical Pitchfork Bifurcation
(e) Supercritical Hopf Bifurcation
Write the system equations as a part of the report.