Synchrony Branching Lemma for Regular Networks

TL;DR

This paper generalizes the Synchrony Branching Lemma for regular networks, establishing conditions for steady-state bifurcations with various synchrony patterns based solely on network structure.

Contribution

It extends the existing lemma to show the generic existence of bifurcation branches with maximal synchrony and provides criteria for submaximal synchrony bifurcations.

Findings

Proves the generic existence of steady-state bifurcation branches with maximal synchrony.

Provides necessary and sufficient conditions for submaximal synchrony bifurcations.

Shows that lattice structure alone does not determine bifurcation support.

Abstract

Coupled cell systems are dynamical systems associated to a network and synchrony subspaces, given by balanced colorings of the network, are invariant subspaces for every coupled cell systems associated to that network. Golubitsky and Lauterbach (SIAM J. Applied Dynamical Systems, 8 (1) 2009, 40-75) prove an analogue of the Equivariant Branching Lemma in the context of regular networks. We generalize this result proving the generic existence of steady-state bifurcation branches for regular networks with maximal synchrony. We also give necessary and sufficient conditions for the existence of steady-state bifurcation branches with some submaximal synchrony. Those conditions only depend on the network structure, but the lattice structure of the balanced colorings is not sufficient to decide which synchrony subspaces support a steady-state bifurcation branch.