Self Organization and Self Avoiding Limit Cycles

A simple periodically driven system displaying rich behavior is introduced and studied. The system self-organizes into a mosaic of static ordered regions with three possible patterns, which are threaded by one-dimensional paths on which a small number of mobile particles travel. These trajectories are self-avoiding and non-intersecting, and their relationship to self-avoiding random walks is explored. Near $\rho=0.5$ the distribution of path lengths becomes power-law like up to some cutoff length, suggesting a possible critical state.