Heterogeneous Van der Pol Oscillators under Strong Coupling

Abstract

The behavior of a network of heterogeneous Van der Pol oscillators under output diffusive coupling is studied. By the intuition on strong coupling given by the series of 'blended dynamics' research, we first show that under sufficiently strong coupling gain the network achieves practical synchronization among heterogeneous oscillators even when there is no common internal model. Then, a further analysis finds a necessary and sufficient condition for the network to achieve synchronous and oscillatory behavior. It should be noted that the condition is the existence of a stable limit cycle for an emergent Van der Pol oscillator we introduce in this paper. Since the stability of each oscillator is not asked, the individual can be any system having the same structure with the Van der Pol oscillator, for instance, double integrator, linear oscillator, and a Van der Pol oscillator with an unstable limit cycle. Finally, by applying a modified singular perturbation theory, we show that the individual agent oscillates near the stable limit cycle of the emergent Van der Pol oscillator.