Dynamics of two coupled van der Pol–Mathieu oscillators

The interaction of two van der Pol–Mathieu oscillators, the simplest models of self-oscillatory systems with parametric excitation, is studied. The dynamics of an individual oscillator is determined either by its self-oscillatory or parametric component, therefore, all possible combinations are considered: both oscillators in a self-oscillatory regime; both oscillators in parametric regime; one oscillator in self-oscillatory regime, another one in parametric regime. To determine the nature of the system dynamics, both complete equations and averaged equations for amplitudes and phases are used, Lyapunov exponents, sections of phase trajectories, and oscillation power spectra are calculated. The behavior of the system under study is compared with the classical case of the interaction of two dissipatively coupled van der Pol oscillators.