American Institute of Mathematical Sciences

September  2013, 18(7): 1827-1844. doi: 10.3934/dcdsb.2013.18.1827

Phase-locked trajectories for dynamical systems on graphs

 1 Department of Mathematics & Center for Dynamics, TU Dresden, 01069 Dresden, Germany, Germany

Received  March 2012 Revised  February 2013 Published  May 2013

We prove a general result on the existence of periodic trajectories of systems of difference equations with finite state space which are phase-locked on certain components which correspond to cycles in the coupling structure. A main tool is the new notion of order-induced graph which is similar in spirit to a Lyapunov function. To develop a coherent theory we introduce the notion of dynamical systems on finite graphs and show that various existing neural networks, threshold networks, reaction-diffusion automata and Boolean monomial dynamical systems can be unified in one parametrized class of dynamical systems on graphs which we call threshold networks with refraction. For an explicit threshold network with refraction and for explicit cyclic automata networks we apply our main result to show the existence of phase-locked solutions on cycles.
Citation: Jeremias Epperlein, Stefan Siegmund. Phase-locked trajectories for dynamical systems on graphs. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1827-1844. doi: 10.3934/dcdsb.2013.18.1827
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