The double Hopf bifurcation with 2:1 resonance

Abstract

Many problems of physical interest involve the nonlinear interaction of two oscillators with different frequencies. When these frequencies are incommensurate, the interaction is straightforward, involving the amplitude rather than the phase of each oscillator. When the frequencies are in a ratio closely corresponding to a rational fraction of small denominator, phase locking occurs and the dynamics is much richer. In this paper we investigate fully the important case where the frequencies are in the ratio 2:1. The analysis allows for small departures from this ratio and treats not only the generic problem for small amplitudes but also an important degenerate case which appears to capture most of the dynamics at larger amplitudes.