Phase-locked trajectories for dynamical systems on graphs

Abstract

We prove a general result on the existence of periodic trajectories of systemsof difference equations with finite state space which are phase-locked oncertain components which correspond to cycles in the coupling structure. A maintool is the new notion of order-induced graph which is similar in spirit to aLyapunov function. To develop a coherent theory we introduce the notion ofdynamical systems on finite graphs and show that various existing neuralnetworks, threshold networks, reaction-diffusion automata and Boolean monomialdynamical systems can be unified in one parametrized class of dynamical systemson graphs which we call threshold networks with refraction. For an explicitthreshold network with refraction and for explicit cyclic automata networkswe apply our main result to show the existenceof phase-locked solutions on cycles.