2018/01/01 by Sören Schwenker · 6 citations
Computer Science · Engineering · Mathematics · Physics and Astronomy · #Bifurcation #Chaos control and synchronization #Control and Stability of Dynamical Systems #Dynamical systems theory #Equivariant map #Generalization #Homogeneous #Homogeneous space #Indecomposable module #Monoid #Nonlinear Dynamics and Pattern Formation #math.DS #msc:20M30 #msc:37G10 #msc:37G40 #msc:37H20
paper · pdf · doi:10.1137/17m116118x
published in SIAM Journal on Mathematical Analysis 50(3), 2466-2485 (Society for Industrial and Applied Mathematics) · 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
openalex publication_date 2018/01/01 · openalex created_date 2018/03/06 · arxiv created 2018/10/09 · arxiv updated 2018/10/10 · openalex updated_date 2026/08/05
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, SIAM J. Math. Anal., 46 (2014), pp. 1577--1609]. 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 noncompact 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.