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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.

In honor of Marty Golubitsky on the occasion of his seventieth birthday. AbstractIn this paper we continue the study of group representations which are counterexamples to the Ize conjecture. As in the previous papers by Lauterbach [14] and Lauterbach and Matthews [15] we find new infinite series of finite groups leading to such counterexamples. These new series are quite different from the previous ones, for example the group orders do not form an arithmetic progression. However, as before we find Lie groups which contain all these groups. This additional structure was observed, but not used in the previous studies of this problem. Here we also investigate the related bifurcations. To a large extent, these are closely related to the presence of mentioned compact Lie group containing the finite groups. This might give a tool to study the bifurcations related to all low dimensional counterexamples of the Ize conjecture. It also gives an indication of where we can expect to find examples where the bifurcation behavior is different from what we have seen in the known examples.

The authors of Berg et al. [J. Algebra 348 (2011) 446–461] provide an algorithm for finding a complete system of primitive orthogonal idempotents for [Formula: see text], where [Formula: see text] is any finite [Formula: see text]-trivial monoid. Their method relies on a technical result stating that [Formula: see text]-trivial monoid are equivalent to so-called weakly ordered monoids. We provide an alternative algorithm, based only on the simple observation that an [Formula: see text]-trivial monoid may be realized by upper triangular matrices. This approach is inspired by results in the field of coupled cell network dynamical systems, where [Formula: see text]-trivial monoids (the opposite notion) correspond to so-called feed-forward networks. We first show that our algorithm works for [Formula: see text], after which we prove that it also works for [Formula: see text] where [Formula: see text] is an arbitrary ring with a known complete system of primitive orthogonal idempotents. In particular, our algorithm works if [Formula: see text] is any field. In this respect our result constitutes a considerable generalization of the results in Berg et al. [J. Algebra 348 (2011) 446–461]. Moreover, the system of idempotents for [Formula: see text] is obtained from the one our algorithm yields for [Formula: see text] in a straightforward manner. In other words, for any finite [Formula: see text]-trivial monoid [Formula: see text] our algorithm only has to be performed for [Formula: see text], after which a system of idempotents follows for any ring with a given system of idempotents.