Phase-locked trajectories for dynamical systems on graphs

Jeremias Epperlein; Stefan Siegmund

doi:10.3934/dcdsb.2013.18.1827

Article Contents

2013, Volume 18, Issue 7: 1827-1844. Doi: 10.3934/dcdsb.2013.18.1827

This issue Previous Article Effects of white noise in multistable dynamics Next Article Finite element approximations for a linear Cahn-Hilliard-Cook equation driven by the space derivative of a space-time white noise

Phase-locked trajectories for dynamical systems on graphs

Jeremias Epperlein^1, and
Stefan Siegmund^1,

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

Received: February 29, 2012

Revised: January 31, 2013

Published: August 2013

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Abstract

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.

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Mathematics Subject Classification: Primary: 37B15, 05C20, 37B25; Secondary: 92C20.

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