Cycling Behavior in Near-Identical Cell Systems

Abstract

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.