Chaos in coupled heteroclinic cycles and its piecewise-constant representation

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