Math 4556 Autumn 2022 Problem Set 3
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Math 4556
PRoBLEM SET 3
Autumn 2022
1. Imperfect transcritical bifurcation ( #3.6.2)
Consider the system x˙ = h+Tx − x2 . When h = 0, this system undergoes a transcritical bifurcation at T = 0. Our goal is to see how the bifurcation diagram of x* vs. T is affected by the imperfection parameter h.
a) Plot the bifurcation diagram for x˙ = h + Tx − x2 , for h < 0, h = 0, and h > 0.
b) Sketch the regions in the (T, h) plane that correspond to qualitatively different vector fields, and identify the bifurcations that occur on the boundaries of those regions.
Note: on this problem set, it is not necessary to transform the system into normal form to confirm the type of bifurcation. You may conclude simply based on the vector fields and the bifurcation diagram or use instead the partial derivative criteria presented in class.
2. #3.1.2
3. #3.1.3
4. #3.1.4
5. #3.1.5
2023-02-07