Chaos in coupled heteroclinic cycles and its piecewise-constant representation

Abstract

We consider two robust heteroclinic cycles rotating in opposite directions, coupled by diffusive terms. A complete synchronization is impossible in this system. Numerical studies show that chaos is abundant at low levels of coupling. With increasing coupling strength, several symmetry-changing transitions are observed, and finally, a stable periodic regime appears via an inverse period-doubling cascade. To reveal the behavior at extremely small couplings, a piecewise constant model for the dynamics is proposed. Within this model, we numerically construct a Poincaré map for a chaotic state, which appears to be an expanding non-invertible circle map, thus confirming the abundance of chaos in the small coupling limit. We also show that within the piecewise constant description, there is a set of periodic solutions with different phase shifts between subsystems due to dead zones in the coupling.