Dynamics of a delayed van der Pol-Mathieu oscillator

摘要

The dynamics of a van der Pol-Mathieu (vdPM) equation with time delay is considered. The vdPM model can be realized as the governing equation for dust density in a simplified model of dusty plasma. The dynamics of the time-delayed equation is analyzed by separating the timescales of the system assuming that the fundamental simple harmonic oscillator is at least an O(ε) dominating other terms of the oscillator including time-delay, where ε1. Our analytic prediction of the slow-flow system correctly represents the dynamics of the original system, showing periodic creation and annihilation of multi-periodic limit cycles. The original system is then analyzed using the DDE-Biftool [1] bifurcation analysis tool. We show that for large time-delay, the system undergoes a double-Hopf bifurcation, whereas for small delay, it undergoes a Bogdanov-Takens bifurcation.