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arXiv · 1802.08490

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

Sören Schwenker
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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.

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Sören Schwenker. 2018-02-23. Generic steady state bifurcations in monoid equivariant dynamics with applications in homogeneous coupled cell systems. https://doi.org/10.1137/17m116118x

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