Synchrony Branching Lemma for Regular Networks

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.

Steady-state bifurcations for three-cell networks with asymmetric inputs

“…We start by addressing the simplest case where µ is simple and the synchrony space ∆ is maximal, which is a common case, specially when the synchrony subspace ∆ has a low dimension. Similar cases have been studied in [23,20] for regular networks and our approach is similar. The idea is to apply Lyapunov-Schmidt Reduction [13], reducing the steady-state bifurcation problem to a one-dimensional steady-state bifurcation problem, since the eigenfunction µ is simple.…”

“…. , ỹm λ, λ) = 0 such that ỹ1 [20,Proposition 3.6], we see that each such solution corresponds to an unique solution of g = 0 and a unique bifurcation branch of f N . Proposition 3.3.…”

Reduced Lattices of Synchrony Subspaces and Their Indices

“…In relation to the literature [1,4,10,20], in future work we aim to employ the algorithm for index-based bifurcation analysis, which can be used for predicting multiple bifurcating branches from a single bifurcation point in regular coupled cell networks. It will be especially interesting to investigate some counter examples for which our algorithm fails, and to examine what our postulated reduction can tell us about generic synchrony-breaking bifurcations.…”

“…Dynamics and bifurcations of coupled cell systems have been much studied, for example Leite et al [16], Elmhirst et al [7], Aguiar et al [4], Golubitsky et al [10], Stewart et al [22] and Soares [20] for regular networks; and Stewart et al [23], Golubitsky et al [12] and Gandhi et al [9] and Aguiar et al [1] for more general coupled cell.…”

“…This enables us to visualise a finite lattice of synchrony subspaces as a combination of an infinite lattice of invariant subspaces with a finite partition lattice. Construction and properties of a lattice of synchrony subspaces have been studied [1,2,13,15,18,19,20,21]. When the adjacency matrix of a given regular network has non-simple eigenvalues, the size of a lattice of synchrony subspaces increases due to higher dimension eigenspaces.…”