Bifurcations in dynamical systems with interior symmetry

Bifurcations in dynamical systems with interior symmetry

Author Antoneli, Fernando Google Scholar
Institution Universidade Federal de São Paulo (UNIFESP)
Abstract We propose a definition of interior symmetry in the context of general dynamical systems. This concept appeared originally in the theory of coupled cell networks, as a generalization of the idea of symmetry of a network. the notion of interior symmetry introduced here can be seen as a special form of forced symmetry breaking of an equivariant system of differential equations. Indeed, we show that a dynamical system with interior symmetry can be written as the sum of an equivariant system and a 'perturbation term' which completely breaks the symmetry. Nonetheless, the resulting dynamical system still retains an important feature common to systems with symmetry, namely, the existence of flow-invariant subspaces. We define interior symmetry breaking bifurcations in analogy with the definition of symmetry breaking bifurcation from equivariant bifurcation theory and study the codimension one steady-state and Hopf bifurcations. Our main result is the full analogues of the well-known Equivariant Branching Lemma and the Equivariant Hopf Theorem from the bifurcation theory of equivariant dynamical systems in the context of interior symmetry breaking bifurcations.
Keywords equivariant dynamical systems
symmetry breaking bifurcation
equivariant branching lemma
equivariant Hopf theorem
compact Lie group
Language English
Date 2010-01-01
Published in Dynamical Systems-an International Journal. Abingdon: Taylor & Francis Ltd, v. 25, n. 2, p. 239-251, 2010.
ISSN 1468-9367 (Sherpa/Romeo, impact factor)
Publisher Taylor & Francis Ltd
Extent 239-251
Access rights Closed access
Type Article
Web of Science ID WOS:000277742000006

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