Chaos in Coupled Heteroclinic Cycles and its Piecewise-Constant Representation

TL;DR

This paper investigates chaos in coupled heteroclinic cycles, revealing how chaos persists at low coupling, transitions to periodic orbits at higher coupling, and introduces a piecewise-constant model to analyze small coupling behavior.

Contribution

It introduces a novel piecewise-constant model to analyze chaos and symmetry transitions in coupled heteroclinic cycles, providing new insights into their dynamics at small coupling.

Findings

Chaos is abundant at low coupling levels.

Symmetry-changing transitions occur as coupling increases.

A stable periodic orbit emerges via inverse period-doubling cascade.

Abstract

We consider two stable heteroclinic cycles rotating in opposite directions, coupled via diffusive terms. A complete synchronization in this system is impossible, and numerical exploration shows that chaos is abundant at low levels of coupling. With increase of coupling strength, several symmetry-changing transitions are observed, and finally a stable periodic orbit appears via an inverse period-doubling cascade. To reveal the behavior at extremely small couplings, a piecewise-constant model for the dynamics is suggested. Within this model we construct a Poincar\'e map for a chaotic state numerically, it appears to be an expanding non-invertable circle map thus confirming 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.