Generic steady state bifurcations in monoid equivariant dynamics with applications in homogeneous coupled cell systems

TL;DR

This paper generalizes the understanding of steady state bifurcations in symmetric dynamical systems to include monoid symmetries, extending classical results from group symmetries to more general algebraic structures, with applications in coupled cell networks.

Contribution

It proves that steady state bifurcations in monoid-equivariant systems typically occur along absolutely indecomposable subrepresentations, broadening the scope beyond group symmetries.

Findings

Bifurcations occur along absolutely indecomposable subrepresentations in monoid symmetric systems.

The results include non-compact symmetry groups.

Applications to bifurcation theory of homogeneous coupled cell networks.

Abstract

We prove that steady state bifurcations in finite-dimensional dynamical systems that are symmetric with respect to a monoid representation generically occur along an absolutely indecomposable subrepresentation. This is stated as a conjecture in B. Rink and J. Sanders, "Coupled cell networks and their hidden symmetries", SIAM J. Math. Anal., 46 (2014). It is a generalization of the well-known fact that generic steady state bifurcations in equivariant dynamical systems occur along an absolutely irreducible subrepresentation if the symmetries form a group - finite or compact Lie. Our generalization also includes non-compact symmetry groups. The result has applications in bifurcation theory of homogeneous coupled cell networks as they can be embedded (under mild additional assumptions) into monoid equivariant systems.