Phase-locked trajectories for dynamical systems on graphs

Jeremias Epperlein, Stefan Siegmund

Research output: Contribution to journalArticlepeer-review

3 Scopus citations


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.

Original languageEnglish
Pages (from-to)1827-1844
Number of pages18
JournalDiscrete and Continuous Dynamical Systems - Series B
Issue number7
StatePublished - 1 Sep 2013
Externally publishedYes


  • Directed graph
  • Discrete dynamical system
  • Neural network
  • Partial order
  • Phase-locking
  • Synchronization

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Applied Mathematics


Dive into the research topics of 'Phase-locked trajectories for dynamical systems on graphs'. Together they form a unique fingerprint.

Cite this