Narrow Results

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 one-dimensional 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.

A generic pattern of collective behavior of symmetric networks of coupled identical cells is cycling behavior. In networks modeled by symmetric systems of differential (difference) equations, cycling behavior appears in the form of solution trajectories (orbits) that linger around symmetrically related steady-states (fixed points) or periodic solutions (orbits) or even chaotic attractors. In this last case, it leads to what is called "cycling chaos". In particular, Dellnitz et al. [1995] demonstrated the existence of cycling chaos in continuous-time three-cell systems modeled by Chua's circuit equations and Lorenz equations, while Palacios [2002, 2003] later demonstrated the existence of cycling chaos in discrete-time cell systems. In this work, we consider two issues that follow-up from these previous works. First of all, we address the generalization of existence of cycling behavior in continuous-time systems with more than three cells. We demonstrate that increasing the number of cells, while maintaining the same network connectivity used by Dellnitz et al. [1995], is not enough to sustain the nature of a cycle, in which only one cell is active at any given time. Secondly, we address the existence of cycling behavior in networks with near-identical cells, where the internal dynamics of each cell is governed by an identical model equation but with possibly different parameter values. We show that, under a new connectivity scheme, cycling behavior can also occur in networks with near-identical cells.

Recently, we looked more closely into the Rabinovich–Fabrikant system, after a decade of study [Danca & Chen, 2004], discovering some new characteristics such as cycling chaos, transient chaos, chaotic hidden attractors and a new kind of saddle-like attractor. In addition to extensive and accurate numerical analysis, on the assumptive existence of heteroclinic orbits, we provide a few of their approximations.