Math 5110/6830
Instructor: Alla Borisyuk
Homework 8.2
Due: November 3
1. (St: ch3) For each of the following exercises do:
(i) sketch all the qualitatively dierent vector elds that occur as is varied.
(ii) Find the bifurcation value of (iii) Identify the bifurcation.
(iv) Sketch the bifurcation diagram with as a parameter.
a)x_ = 1 + rx + x2
b)x_ = x rx(1 x)
c)x_ = x(r ex )
d)x_ = x + 4x3
2. To identify a hopf bifurcation analytically the following criterion is used:
the eigenvalues at the bifurcation value are purely imaginary. That means
that on one side of the bifurcation the xed point is stable and on the other unstable, but in both cases it has trajectories oscillating around it. By doing
the linearization at the origin and computing the eigenvalues, show that the
system x_ = y + x + xy2 , y_ = x + y x2 undergoes a Hopf bifurcation at
= 0.
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