Generic Steady State Bifurcations in Monoid Equivariant Dynamics with Applications in Homogeneous Coupled Cell Systems

S. Schwenker, SIAM Journal on Mathematical Analysis 50 (2018) 2466–2485.

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Journal Article | Published | English
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, 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.
Publishing Year
Journal Title
SIAM Journal on Mathematical Analysis
Volume
50
Issue
3
Page
2466-2485
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Schwenker S. Generic Steady State Bifurcations in Monoid Equivariant Dynamics with Applications in Homogeneous Coupled Cell Systems. SIAM Journal on Mathematical Analysis. 2018;50(3):2466-2485. doi:10.1137/17m116118x
Schwenker, S. (2018). Generic Steady State Bifurcations in Monoid Equivariant Dynamics with Applications in Homogeneous Coupled Cell Systems. SIAM Journal on Mathematical Analysis, 50(3), 2466–2485. https://doi.org/10.1137/17m116118x
@article{Schwenker_2018, title={Generic Steady State Bifurcations in Monoid Equivariant Dynamics with Applications in Homogeneous Coupled Cell Systems}, volume={50}, DOI={10.1137/17m116118x}, number={3}, journal={SIAM Journal on Mathematical Analysis}, publisher={Society for Industrial & Applied Mathematics (SIAM)}, author={Schwenker, Sören}, year={2018}, pages={2466–2485} }
Schwenker, Sören. “Generic Steady State Bifurcations in Monoid Equivariant Dynamics with Applications in Homogeneous Coupled Cell Systems.” SIAM Journal on Mathematical Analysis 50, no. 3 (2018): 2466–85. https://doi.org/10.1137/17m116118x.
S. Schwenker, “Generic Steady State Bifurcations in Monoid Equivariant Dynamics with Applications in Homogeneous Coupled Cell Systems,” SIAM Journal on Mathematical Analysis, vol. 50, no. 3, pp. 2466–2485, 2018, doi: 10.1137/17m116118x.
Schwenker, Sören. “Generic Steady State Bifurcations in Monoid Equivariant Dynamics with Applications in Homogeneous Coupled Cell Systems.” SIAM Journal on Mathematical Analysis, vol. 50, no. 3, Society for Industrial & Applied Mathematics (SIAM), 2018, pp. 2466–85, doi:10.1137/17m116118x.

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