Heteroclinic Cycles in Coupled Systems of Difference Equations

Abstract

Cycling behavior, in which solution trajectories linger around steady-states and periodic solutions, is known to be a generic feature of coupled cell systems of differential equations. In this type of systems, cycling behavior can even occur independently of the internal dynamics of each cell. This conclusion has lead to the discovery of "cycling chaos", in which solution trajectories cycle around symmetrically related chaotic sets. In this work, we demonstrate that cycling behavior also occurs in coupled systems of difference equations. More specifically, we prove the existence of structurally stable cycles between fixed points, and use numerical simulations to illustrate that the resulting cycles can also persist independently of the internal dynamics of each cell. Consequently, we demonstrate that cycles involving periodic orbits as well as cycling chaos also occur in systems of difference equations.