Math 4556 Problem Set 2 Autumn 2022
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Math 4556
PRoBLEM SET 2
Autumn 2022
For problems 1 and 2, also find new variables X and R so that the system reduces to the approx-
imate normal form
X˙ = RX − X2 + (higher order terms)
(See Example 3.2.2 on page 53 of the text.)
1. (#3.2.2)
2. (#3.2.4)
3. (#3.2.5)
4. (#3.4.1)
5. (#3.4.2)
6. (#3.4.3)
7. Distinguishing types of bifurcations; analyzing transcritical and pitchfork bifurcations
Each of the following equations has a bifurcation as the parameter r is varied. If the bifurcation type is transcritical or pitchfork, find the value of r at which the bifurcation occurs, classify the bifurcation as transcritical, supercritical pitchfork or subcritical pitchfork, and sketch the bifurcation diagram of fixed points x* vs. r . If the bifurcation type is not transcritical or pitchfork, just say how you know that it isn’t transcritical or pitchfork (you do not need to analyze those).
a) x˙ = rx −
b) x˙ = rx −
c) x˙ = x + tanh(rx)
d) x˙ = rx +
2023-02-04