Synchrony and antisynchrony for difference-coupled vector fields on graph network systems

Research output: Contribution to journalArticle

Abstract

We define a graph network to be a coupled cell network where there are only one type of cell and one type of symmetric coupling between the cells. For a difference-coupled vector field on a graph network system, all the cells have the same internal dynamics, and the coupling between cells is identical, is symmetric, and depends only on the difference of the states of the interacting cells. We define four nested sets of difference-coupled vector fields by adding further restrictions on the internal dynamics and the coupling functions. These restrictions require that these functions preserve zero or are odd or linear. We characterize the synchrony and antisynchrony subspaces with respect to these four subsets of admissible vector fields. Synchrony and antisynchrony subspaces are determined by partitions and matched partitions of the cells that satisfy certain balance conditions. We compute the lattice of synchrony and antisynchrony subspaces for some examples of graph networks. We also apply our theory to systems of coupled van der Pol oscillators.

Original languageEnglish (US)
Pages (from-to)904-938
Number of pages35
JournalSIAM Journal on Applied Dynamical Systems
Volume18
Issue number2
DOIs
StatePublished - Jan 1 2019

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Synchrony
Vector Field
Cell
Graph in graph theory
Subspace
Partition
Restriction
Internal
Van Der Pol Oscillator
Coupled Oscillators
Odd
Subset
Zero

Keywords

  • Bifurcation
  • Coupled systems
  • Synchrony

ASJC Scopus subject areas

  • Analysis
  • Modeling and Simulation

Cite this

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title = "Synchrony and antisynchrony for difference-coupled vector fields on graph network systems",
abstract = "We define a graph network to be a coupled cell network where there are only one type of cell and one type of symmetric coupling between the cells. For a difference-coupled vector field on a graph network system, all the cells have the same internal dynamics, and the coupling between cells is identical, is symmetric, and depends only on the difference of the states of the interacting cells. We define four nested sets of difference-coupled vector fields by adding further restrictions on the internal dynamics and the coupling functions. These restrictions require that these functions preserve zero or are odd or linear. We characterize the synchrony and antisynchrony subspaces with respect to these four subsets of admissible vector fields. Synchrony and antisynchrony subspaces are determined by partitions and matched partitions of the cells that satisfy certain balance conditions. We compute the lattice of synchrony and antisynchrony subspaces for some examples of graph networks. We also apply our theory to systems of coupled van der Pol oscillators.",
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N2 - We define a graph network to be a coupled cell network where there are only one type of cell and one type of symmetric coupling between the cells. For a difference-coupled vector field on a graph network system, all the cells have the same internal dynamics, and the coupling between cells is identical, is symmetric, and depends only on the difference of the states of the interacting cells. We define four nested sets of difference-coupled vector fields by adding further restrictions on the internal dynamics and the coupling functions. These restrictions require that these functions preserve zero or are odd or linear. We characterize the synchrony and antisynchrony subspaces with respect to these four subsets of admissible vector fields. Synchrony and antisynchrony subspaces are determined by partitions and matched partitions of the cells that satisfy certain balance conditions. We compute the lattice of synchrony and antisynchrony subspaces for some examples of graph networks. We also apply our theory to systems of coupled van der Pol oscillators.

AB - We define a graph network to be a coupled cell network where there are only one type of cell and one type of symmetric coupling between the cells. For a difference-coupled vector field on a graph network system, all the cells have the same internal dynamics, and the coupling between cells is identical, is symmetric, and depends only on the difference of the states of the interacting cells. We define four nested sets of difference-coupled vector fields by adding further restrictions on the internal dynamics and the coupling functions. These restrictions require that these functions preserve zero or are odd or linear. We characterize the synchrony and antisynchrony subspaces with respect to these four subsets of admissible vector fields. Synchrony and antisynchrony subspaces are determined by partitions and matched partitions of the cells that satisfy certain balance conditions. We compute the lattice of synchrony and antisynchrony subspaces for some examples of graph networks. We also apply our theory to systems of coupled van der Pol oscillators.

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