Generic steady state bifurcations in monoid equivariant dynamics with applications in homogeneous coupled cell systems
arXiv:1802.08490 · doi:10.1137/17M116118X
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.
19, pages, 1 figure; Minor revisions in the example: cosmetic changes in Figure 1 and its explanation on page 14, replaced "saddle-node bifurcation" with "transcritical bifurcation" on page 15