Ordinary Differential Equations and Dynamical Systems Vol2 · Problem 8.2

Center Manifolds and Bifurcation Theory

Mathematics — Calculus · periodic-solutions-planar-flow

Problem

Let

h : [0,\infty) \to (0,\infty)

C^1

. For

x = (x_1, x_2) \in \mathbb{R}^2

, let

r = \|x\|

. For

p \in \mathbb{R}^2

, let

x(t, p)

denote the solution of the IVP

x_1' = h(r^2)x_2, \quad x_2' = -h(r^2)x_1 \quad x(0, p) = p.

(a) Show that for every

p \in \mathbb{R}^2

and

t \in \mathbb{R}

\|x(t, p)\| = \|p\|

. (b) Show that

x(t, p)

is periodic with period

2\pi/h(\|p\|^2)

, for

p \neq 0

. (c) If

h

is strictly increasing, show that for each

p

x(t, p)

is orbitally stable, but not stable.