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

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