Generic steady state bifurcations in monoid equivariant dynamics with applications in homogeneous coupled cell systems
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.
Cite
@article{arxiv.1802.08490,
title = {Generic steady state bifurcations in monoid equivariant dynamics with applications in homogeneous coupled cell systems},
author = {Sören Schwenker},
journal= {arXiv preprint arXiv:1802.08490},
year = {2018}
}
Comments
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