## Description

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

(a) Saddle Node Bifurcation

(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.