Classical Mechanics (Kibble & Berkshire) · Problem 8.9

The energy principle — T.W.B. Kibble, F.H. Berkshire

Physics — Mechanics · phase-paths-limit-cycle-nonlinear

Problem

A non-linear oscillator satisfies the equation

\ddot{x} + \left(x^2 + \dot{x}^2 - 1\right)\dot{x} + x = 0.

Find the polar equations for the motion of the phase points. Show that any phase path that starts in the domain

1 < r < \sqrt{3}

spirals clockwise and tends to the limit cycle

r = 1

. [The same is true of phase paths that start in the domain

0 < r < 1

.] What is the period of the limit cycle?