Chaotic motion of a weakly nonlinear, modulated oscillator.

A weakly nonlinear oscillator of natural frequency omega(0) and damping ratio delta is driven by an amplitude-modulated force of dimensionless amplitude epsilon, carrier frequency omega, and modulation frequency omega(1). Each of (omega - omega(0))/omega(0), omega(1)/omega(0), and delta is O(epsilon(2/3)) as epsilon --> 0. The response in this resonant neighborhood is a slowly modulated sine wave, the envelope of which is described by three first-order, autonomous, ordinary differential equations. This envelope, which is periodic for delta >> epsilon(2/3), is chaotic in certain ranges of (omega - omega(0))/omega(0) and omega(1)/omega(0) if delta/epsilon(2/3) is sufficiently small.