Cycling Chaos in One-Dimensional Coupled Iterated Maps

Cycling behavior involving steady-states and periodic solutions is known to be a generic feature of continuous dynamical systems with symmetry. Using Chua’s circuit equations and Lorenz equations, Dellnitz et al. [1995] showed that “cycling chaos”, in which solution trajectories cycle around symmetrically related chaotic sets, can also be found generically in coupled cell systems of differential equations with symmetry. In this work, we use numerical simulations to demonstrate that cycling chaos also occurs in discrete dynamical systems modeled by onedimensional maps. Using the cubic map f(x, λ) = λx−x3 and the standard logistic map, we show that coupled iterated maps can exhibit cycles connecting fixed points with fixed points and periodic orbits with periodic orbits, where the period can be arbitrarily high. As in the case of coupled cell systems of differential equations, we show that cycling behavior can also be a feature of the global dynamics of coupled iterated maps, which exists independently of the internal dynamics of each map.