Coupled cell networks and their hidden symmetries

TL;DR

This paper reveals that homogeneous coupled cell networks possess hidden symmetries that explain their synchronous solutions, spectral degeneracies, and bifurcation behaviors, providing a classification of bifurcations in small networks.

Contribution

It demonstrates that all homogeneous network dynamical systems have hidden semigroup symmetries, linking spectral degeneracies and bifurcations to indecomposable representations.

Findings

Hidden symmetries explain synchrony and spectral degeneracies.

Bifurcations occur along indecomposable subrepresentations.

Classification of bifurcations in small monoid networks.

Abstract

Dynamical systems with a coupled cell network structure can display synchronous solutions, spectral degeneracies and anomalous bifurcation behavior. We explain these phenomena here for homogeneous networks, by showing that every homogeneous network dynamical system admits a semigroup of hidden symmetries. The synchronous solutions lie in the symmetry spaces of this semigroup and the spectral degeneracies of the network are determined by its indecomposable representations. Under a condition on the semigroup representation, we prove that a one-parameter synchrony breaking steady state bifurcation in a coupled cell network must generically occur along an absolutely indecomposable subrepresentation. We conclude with a classification of generic one-parameter bifurcations in monoid networks with two or three cells.