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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
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: |
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| 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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