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The Structure of Symmetric Attractors
 ARCH. RATIONAL MECH. ANAL
, 1992
"... We consider discrete equivariant dynamical systems and obtain results about the structure of attractors for such systems. We show, for example, that the symmetry of an attractor cannot, in general, be an arbitrary subgroup of the group of symmetries. In addition, there are grouptheoretic restri ..."
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Cited by 25 (10 self)
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We consider discrete equivariant dynamical systems and obtain results about the structure of attractors for such systems. We show, for example, that the symmetry of an attractor cannot, in general, be an arbitrary subgroup of the group of symmetries. In addition, there are grouptheoretic restrictions on the symmetry of connected components of a symmetric attractor. Our methods are topological in nature and exploit connectedness properties of the ambient space. In particular, we prove a general lemma about connected components of the complement of preimage sets and how they are permuted by the mapping. These methods do not themselves depend on equivariance. For example, we use them to prove that the presence of periodic points in the dynamics limits the number of connected components of an attractor, and, for onedimensional mappings, to prove results on sensitive dependence and the density of periodic points.
Symmetric attractors for diffeomorphisms and flows
 Proc. London Math. Soc
, 1996
"... Abstract. Let Γ ⊂ O(n) be a finite group acting onR n. In this work we describe the possible symmetry groups that can occur for attractors of smooth (invertible) Γequivariant dynamical systems. In case R n contains no reflection planes and n ≥ 3, our results imply there are no restrictions on symmm ..."
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Cited by 15 (9 self)
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Abstract. Let Γ ⊂ O(n) be a finite group acting onR n. In this work we describe the possible symmetry groups that can occur for attractors of smooth (invertible) Γequivariant dynamical systems. In case R n contains no reflection planes and n ≥ 3, our results imply there are no restrictions on symmmetry groups. In case n ≥ 4 (diffeomorphisms) and n ≥ 5 (flows), we show that we may construct attractors which are Axiom A. We also give a complete description of what can happen in low dimensions.
Symmetry Breaking Bifurcations of Chaotic Attractors
 INT. J. BIF. CHAOS
, 1994
"... In an array of coupled oscillators synchronous chaos may occur in the sense that all the oscillators behave identically although the corresponding motion is chaotic. When a parameter is varied this fully symmetric dynamical state can lose its stability, and the main purpose of this paper is to inves ..."
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Cited by 7 (4 self)
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In an array of coupled oscillators synchronous chaos may occur in the sense that all the oscillators behave identically although the corresponding motion is chaotic. When a parameter is varied this fully symmetric dynamical state can lose its stability, and the main purpose of this paper is to investigate which type of dynamical behavior is expected to be observed once the loss of stability has occurred. The essential tool is a classification of Lyapunov exponents based on the symmetry of the underlying problem. This classification is crucial in the derivation of the analytical results but it also allows an efficient computation of the dominant Lyapunov exponent associated with each symmetry type. We show how these dominant exponents determine the stability of invariant sets possessing various instantaneous symmetries and this leads to the idea of symmetry breaking bifurcations of chaotic attractors. Finally the results and ideas are illustrated for several systems of coupled oscillators.
Generalizations of a result on symmetry groups of attractors
 in Pattern Formation: Symmetry Methods and Applications
, 1996
"... Abstract. The admissible symmetry groups of attractors for continuous equivariant mappings were classified in Ashwin and Melbourne [1994] and Melbourne, Dellnitz and Golubitsky [1993]. We consider extensions of these results to include attractors in fixedpoint subspaces, attractors for equivarian ..."
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Cited by 7 (3 self)
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Abstract. The admissible symmetry groups of attractors for continuous equivariant mappings were classified in Ashwin and Melbourne [1994] and Melbourne, Dellnitz and Golubitsky [1993]. We consider extensions of these results to include attractors in fixedpoint subspaces, attractors for equivariant diffeomorphisms and flows, and attractors in the presence of a continuous symmetry group. Our results lead to surprising (if somewhat speculative) implications for both theory and applications of equivariant dynamical systems. 1.
On Symmetric Attractors in Reversible Dynamical Systems
, 1997
"... Let \Gamma ae O(n) be a finite group acting orthogonally on IR n . We say that \Gamma is a reversing symmetry group of a homeomorphism, diffeomorphism or flow f t : IR n 7! IR n (t 2 Z Z or t 2 IR) if \Gamma has an index two subgroup ~ \Gamma whose elements commute with f t and for all e ..."
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Cited by 2 (1 self)
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Let \Gamma ae O(n) be a finite group acting orthogonally on IR n . We say that \Gamma is a reversing symmetry group of a homeomorphism, diffeomorphism or flow f t : IR n 7! IR n (t 2 Z Z or t 2 IR) if \Gamma has an index two subgroup ~ \Gamma whose elements commute with f t and for all elements ae 2 \Gamma \Gamma ~ \Gamma and all t , f t ffi ae(x) = ae ffi f \Gammat (x). We give necessary group and representation theoretic conditions for subgroups of reversing symmetry groups to occur as symmetry groups of attractors (Liapunov stable !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 re...
Symmetries of periodic solutions for planar potential systems
 Proc.Am.Math.Soc
, 1996
"... Abstract. In this article we discuss the symmetries of periodic solutions to Hamiltonian systems with two degrees of freedom in mechanical form. The possible symmetries of such periodic trajectories are generated by spatial symmetries (a finite subgroup of O(2)), phaseshift symmetries (the circle ..."
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Abstract. In this article we discuss the symmetries of periodic solutions to Hamiltonian systems with two degrees of freedom in mechanical form. The possible symmetries of such periodic trajectories are generated by spatial symmetries (a finite subgroup of O(2)), phaseshift symmetries (the circle group S 1 ), and a timereversing symmetry (associated with mechanical form). We focus on the symmetries and structures of the trajectories in configuration space (R 2 ), showing that special properties such as selfintersections and brake orbits are consequences of symmetry.
unknown title
, 1995
"... Symmetries of the asymptotic dynamics of random compositions of equivariant maps. ..."
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Symmetries of the asymptotic dynamics of random compositions of equivariant maps.
Lectures on Symmetric Attractors 3: Stable Ergodicity and. . .
"... Pugh & Shub [42] extended this result to general manifolds of constant negative curvature and Wilkinson [44] has proved stable ergodicity of the time one map for all negatively curved surfaces. All of these results are difficult to prove on account of the fact that leaves of the center foliatio ..."
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Pugh & Shub [42] extended this result to general manifolds of constant negative curvature and Wilkinson [44] has proved stable ergodicity of the time one map for all negatively curved surfaces. All of these results are difficult to prove on account of the fact that leaves of the center foliation are typically non compact and so verification of ergodicity under perturbation requires delicate estimates. 3.1. Skew extensions. Partially hyperbolic sets arise naturally in the study of diffeomorphisms equivariant with respect to a compact nonfinite Lie group \Gamma. The set of all \Gammaorbits determines a (singular) foliation G of M . If f : M!M is \Gammaequivariant, then G is finvariant. Example 3.1. Suppose that f (\Gammax) = \Gammax. It is easy to show that one can choose a \Gammainvariant Rie
Lectures on Symmetric Attractors
, 1997
"... set up 15 4.2. Geometric realization 15 5. Notes on Lecture 1 17 5.1. Haar measure 17 5.2. Lie groups 18 5.3. Representations and actions 18 5.4. Orbits and isotropy groups 19 5.5. Mappings and isotropy groups 19 5.6. Slice theorem 20 5.7. Isotopy theorem 21 6. Lecture II: Constructing hyperbolic sy ..."
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set up 15 4.2. Geometric realization 15 5. Notes on Lecture 1 17 5.1. Haar measure 17 5.2. Lie groups 18 5.3. Representations and actions 18 5.4. Orbits and isotropy groups 19 5.5. Mappings and isotropy groups 19 5.6. Slice theorem 20 5.7. Isotopy theorem 21 6. Lecture II: Constructing hyperbolic symmetric attractors 22 6.1. An example in R 3 22 7. Notes on Lecture II 26 8. Lecture III: Stable ergodicity of skew extensions 28 8.1. Skew extensions 28 8.2. Result of AdlerKitchensShub 28 8.3. Results of Parry, ParryPollicott 30 8.4. Skew Extensions by general compact connected Lie groups 31 8.5. Sketch proof of Theorem 8.7  \Gamma semisimple 31 8.6. Hyperbolicity for equivariant diffeomorphisms 33 8.7. A nonuniformly hyperbolic base 34 9. Notes on Lecture III 35 References LECTURES ON SYMMETRIC ATTRACTORS 3 Figure 1. Attractor of a planar map with 3fold symmetry 1. Introduction These notes are an expanded version of three lectures given at the DANSE Workshop on Equivariant Dyna...