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Heteroclinic cycles involving periodic solutions in mode interactions with O(2) symmetry

Published online by Cambridge University Press:  14 November 2011

I. Melbourne ,
P. Chossat  and
M. Golubitsky
I. Melbourne
Affiliation:
Department of Mathematics, University of Houston, Houston, TX 77204–3746, U.S.A
P. Chossat
Affiliation:
I.M.S.P., Université de Nice, Pare Valrose, F-06034 Nice Cedex, France
M. Golubitsky
Affiliation:
Department of Mathematics, University of Houston, Houston, TX 77204–3746, U.S.A
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Synopsis

In this paper we show that in O(2) symmetric systems, structurally stable, asymptoticallystable, heteroclinic cycles can be found which connect periodic solutions with steady states and periodic solutions with periodic solutions. These cycles are found in the third-order truncated normal forms of specific codimension two steady-state/Hopf and Hopf/Hopf mode interactions.

We find these cycles using group-theoretic techniques; in particular, we look for certainpatterns in the lattice of isotropy subgroups. Once the pattern has been identified, the heteroclinic cycle can be constructed by decomposing the vector field on fixed-point subspaces into phase/amplitude equations (it is here that we use the assumption of normal form). The final proof of existence (and stability) relies on explicit calculations showing that certain eigenvalue restrictions can be satisfied.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 113 , Issue 3-4 , 1989 , pp. 315 - 345
Copyright
Copyright © Royal Society of Edinburgh 1989

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