MATH 557 Homework Set #11
Fall 2015
44. If µ1 , µ2 have the same sign, show by explicitly giving a homeomorphism that the
flows for x0 = µ1 x and x0 = µ2 x are topologically equivalent.
45. Find any values of µ which are bifurcation points for the system
x01 = µx1 + x2
x02 = x1 − 2x2
46. The ODE
u00 + µ(u2 − 1)u0 + u = 0
containing the parameter µ is known as the Van der Pol equation. For the corresponding
first order system:
a) Discuss the stability and type of the zero solution for any µ.
b) If µ < 0 show that B(0, 1) is a positively invariant set.
c) Show that a Hopf bifurcation takes place at µ = 0.
47. Consider the system
x01 = x32 − 4x1
x02 = x32 − x2 − 3x1
a) Find and classify all critical points.
b) Show that the line x1 = x2 is an invariant set (i.e. both positive and negative
invariant).
c) Is the line x1 = x2 itself an orbit?
d) Show that x1 (t) − x2 (t) → as t → ∞ for any solution. (Suggestion: find an ODE
for x1 − x2 .
48. (Bendixson’s criterion) Let D ⊂ R2 be a simply connected domain (that is to say,
if C is any closed curve in D then the interior of C is also contained in D). If ∇ · f is
not identically zero and does not change sign in D show that x0 = f (x) has no periodic
solution lying in D. (Suggestion: If ψ(t) is a T periodic solution in D,
Z
T
[f1 (ψ(t))ψ20 (t) − f2 (ψ(t))ψ10 (t)] dt = 0
0
View this as a line integral around C, the orbit of ψ, and apply Green’s Theorem.) Using
this result, find conditions on the constants α, β in the second order equation
u00 + αu0 + βu + u2 u0 + u3 = 0
for which Bendixson’s criterion guarantees that there are no periodic solutions.