Math 5110/6830
Homework 9.1
1. For each of the following exercises do:
(i) sketch all qualitatively different vector fields that occur as the parameter is
varied (as in class)
(ii) find the bifurcation value of parameter
(iii) Identify the bifurcation
iv) Sketch the bifurcation diagram
a)ẋ = 1 + rx + x2
b)ẋ = x − rx(1 − x)
c)ẋ = x(r − ex )
d)ẋ = µx + 4x3
2. To identify Hopf bifurcation analytically the following criterion is used:
the eigenvalues at the bifurcation value are purely imaginary. By doing the
linearization at the origin and computing the eigenvalues show that the following
system undergows a Hopf bifurcation at µ = 0:
ẋ = −y + µx + xy 2
ẏ = x + µy − x2
3. Do a change of variables to polar coordinates in a normal form equations
for Hopf bifurcation
4. (Extra credit. Optional for everyone including 6830 students.)
Plot a 3d bifurcation diagram for the Normal form Hopf bifurcation. Play with
light and texture to make it look nice. Send us a file with your picture.
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