Dynamics of two coupled van der Pol-Mathieu oscillators

Abstract

We study the dynamics of two dissipatively coupled van der Pol-Mathieu oscillators. The dynamics of such self-oscillatory systems with parametric excitation is determined either by the self-oscillatory component or by the parametric component depending on the values of the system parameters. This transition is clearly visible on the spectrum of Lyapunov exponents when one of the zero exponents becomes negative. The dynamics of the system is explored for various amplitudes of the parametric excitation in the region of the main parametric resonance. All possible combinations are considered: both oscillators in self-oscillating mode; both oscillators in parametric mode; one oscillator in self-oscillatory mode, the other in parametric mode. The results of a numerical study are presented and a comparison with an analytical solution is shown. The study has shown that in case of weak parametric excitation of both oscillators the dynamics does not differ from one of van der Pol self-oscillators. A strong parametric excitation on one of the oscillators or on both oscillators leads to the appearance on the regime map of phase synchronization regions, parametric and chaotic oscillations. It is also shown that the analytical solution diverges from the numerical one for a large parametric excitation on both oscillators.

Speaker

Ibadulla Ramazanov
Saratov State University
Russia

Discussion

Ask question