Math 2920 – Spring 2011
Homework 4
due Wednesday Feb 16
Problem 1: Write the planar system
(
)(
)(µ − 4(r
x = − y + x µ − (r 2 − 1) 2 µ − 4(r 2 − 1) 2
(
y = x + y µ − (r 2 − 1) 2
2
− 1) 2
)
)
In polar coordinates and show that for µ > 0 the system has one-parameter families of periodic
orbits with radii r = 1 ± µ , 1 ± µ / 2 . Determine the stability of the orbits by computing the
multiplier of the Poincare map. Sketch the bifurcation diagram.
Problem 2: Write the system in polar coordinates. Determine the one-parameter families of
bifurcations of the system. Draw the bifurcation diagram in each case and classify the
bifurcations.
x = − y + xψ (r , µ )
y = x + yψ (r , µ )
(a) ψ (r , µ ) = (r − 1)(r − µ − 1)(r + µ + 1)
(b) ψ (r , µ ) = ( µ − 1)(r 2 − 1)( µ − 1 − (r 2 − 1) 2 )
Problem 3: Find the bifurcation point, sketch the bifurcation diagram and classify the
bifurcation for the following map:
P( x, y, µ ) = ( y,− x / 2 + µy − y 3 )
Problem 4: Show that the system
x = 2 y
y = 2 x − 3 x 2 − y ( x 3 − x 2 + y 2 − ε )
exhibits a homoclinic bifurcation with saddle-node at the origin as ε varies.
(Hint: show that for ε = 0 there is a homoclinic orbit, given by a polynomial in x and y, and that
a change in ε results in a rotation of the field. Use Poincare-Bendixson theorem to argue the
existence of limit cycle.)