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On symmetric attractors in reversible dynamical systems

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Lamb, Jeroen S. W. and Nicol, Matthew (1998) On symmetric attractors in reversible dynamical systems. Physica D: Nonlinear Phenomena, 112 (1-2). pp. 281-297. doi:10.1016/S0167-2789(97)00217-0

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Official URL: http://dx.doi.org/10.1016/S0167-2789(97)00217-0

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Abstract

Let Γ ⊂ O(n) be a finite group acting orthogonally on ℝn. We say that Γ is a reversing symmetry group of a homeomorphism, diffeomorphism or flow ft : ℝn → ℝn (t ∈ ℤ or t ∈ ℝ) if Γ has an index two subgroup Γ̃ whose elements commute with ft and for all elements ρ ∈ Γ - Γ̃ and all t, ft ○ ρ(x) = ρ ○ f-t(x). We give necessary group and representation theoretic conditions for subgroups of reversing symmetry groups to occur as symmetry groups of attractors (Lyapunov stable ω-limit sets). These conditions arise due to topological obstructions. In dimensions 1 and 2 we present a complete description of possible symmetry groups of asymptotically stable attractors for homeomorphisms and diffeomorphisms (these attractors cannot possess reversing symmetries). We also have a fairly complete description in the context of subgroups which contain reversing symmetries. For all dimensions n we present complete results on the possible symmetry groups of connected asymptotically stable attractors. In addition, we summarize our results in the context of reversible equivariant flows, which are complete in dimensions 1 and 2 as well as for subgroups of Γ̃ ⊂ O(n) in any dimension n. A survey of the corresponding results in the equivariant context is also given. Copyright ©1998 Elsevier Science B.V. All rights reserved.

Item Type: Journal Article
Subjects: R Medicine > R Medicine (General)
Divisions: Faculty of Science, Engineering and Medicine > Science > Mathematics
Journal or Publication Title: Physica D: Nonlinear Phenomena
Publisher: Elsevier BV
ISSN: 0167-2789
Official Date: 15 January 1998
Dates:
DateEvent
15 January 1998Published
Volume: 112
Number: 1-2
Page Range: pp. 281-297
DOI: 10.1016/S0167-2789(97)00217-0
Status: Peer Reviewed
Publication Status: Published
Access rights to Published version: Restricted or Subscription Access

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