Synchrony and Anti-Synchrony for Difference-Coupled Vector Fields on Graph Network Systems

TL;DR

This paper characterizes the conditions under which synchrony and anti-synchrony occur in graph network systems with difference-coupled vector fields, providing a framework for understanding their subspace structures and applying it to oscillators.

Contribution

It introduces four nested classes of difference-coupled vector fields and characterizes their associated synchrony and anti-synchrony subspaces using partitions and matched partitions.

Findings

Characterization of synchrony and anti-synchrony subspaces for four classes of vector fields.

Computation of the lattice of subspaces for specific graph networks.

Application of the theory to coupled van der Pol oscillators.

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, 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 anti-synchrony subspaces with respect to these four subsets of admissible vector fields. Synchrony and anti-synchrony subspaces are determined by partitions and matched partitions of the cells that satisfy certain balance conditions. We compute the lattice of synchrony and anti-synchrony subspaces for several graph networks. We also apply our theory to systems of coupled van der Pol oscillators.