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Symmetric patterns in linear arrays of coupled cells (1993)

Epstein, Irving R. ; Golubitsky, Martin

Woodbury, NY : American Institute of Physics (AIP)

Chaos 3 (1993), S. 1-5

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ISSN: 1089-7682

Quelle: AIP Digital Archive

Thema: Physik

Notizen: In this note we show how to find patterned solutions in linear arrays of coupled cells. The solutions are found by embedding the system in a circular array with twice the number of cells. The individual cells have a unique steady state, so that the patterned solutions represent a discrete analog of Turing structures in continuous media. We then use the symmetry of the circular array (and bifurcation from an invariant equilibrium) to identify symmetric solutions of the circular array that restrict to solutions of the original linear array. We apply these abstract results to a system of coupled Brusselators to prove that patterned solutions exist. In addition, we show, in certain instances, that these patterned solutions can be found by numerical integration and hence are presumably asymptotically stable.

Materialart: Digitale Medien

URL: http://dx.doi.org/10.1063/1.165974

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Digitale Medien

The structure of symmetric attractors (1993)

Melbourne, Ian ; Dellnitz, Michael ; Golubitsky, Martin

Springer

Archive for rational mechanics and analysis 123 (1993), S. 75-98

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ISSN: 1432-0673

Quelle: Springer Online Journal Archives 1860-2000

Thema: Mathematik , Physik

Notizen: Abstract 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 group-theoretic restrictions on the symmetry of connected components of a symmetric attractor. The symmetry of attractors has implications for a new type of pattern formation mechanism by which patterns appear in the time-average of a chaotic dynamical system. 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 one-dimensional mappings, to prove results on sensitive dependence and the density of periodic points.