Download x - 2y

MATH 4703
D. Amundsen
Problem Set 4
Due: Apr. 6, 2017
1. Consider the following system
x′ = rx + y − 8x3 + x5
y ′ = x − 2y
(a) Find the fixed points and where they exist for various values of the parameter r.
(b) Linearize and classify each of the fixed points as r varies.
(c) Sketch the bifurcation diagram and indicate the location and type of any bifurcations.
2. Consider the following system
x′ = y − ax
x
y ′ = −by +
1+x
where a, b > 0. Determine the location and type of all bifurcations in terms of the parameters a, b.
3. Consider the following system
x′ = y + µx − x(x2 + y 2 )
y ′ = −x − y(x2 + y 2 )
(a) Convert this system to polar coordinates. i.e. write down equations for r′ , θ′ in terms
of r, θ.
(b) Find and classify all bifurcations in terms of the real parameter µ.
4. A 2-D system of ODEs, written in polar coordinates, takes the form:
r′ = r − r3
θ′ = 2
(a) Determine explicitly the Poincaré Map for this system where the set S is the positive
x axis.
(b) Based on this show that a circular limit cycle exists for a certain radius r = r∗ .
(c) Using the Poincaré Map determine the stability of this limit cycle.
OVER
5. Consider the nonlinear system
x′ = sin (x + 5z)
y ′ = −y + e2z − 1
z ′ = −2y − zex
Show that (0, 0, 0) is a fixed point. Using linearization, sketch the local solution behaviour
around this point, determine the stability and discuss how the limiting behaviours (as
t → ∞) depend on initial condition.