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Asymptotic stability of heteroclinic cycles in systems with symmetry

Published online by Cambridge University Press:  19 September 2008

Martin Krupa  and
Ian Melbourne
Martin Krupa
Affiliation:
Department of Mathematics, University of Groningen, PO Box 800, 9700 AV Groningen, The Netherlands
Ian Melbourne
Affiliation:
Department of Mathematics, University of Houston, Houston, Texas 77204-3476, USA
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Abstract

Systems possessing symmetries often admit heteroclinic cycles that persist under perturbations that respect the symmetry. The asymptotic stability of such cycles has previously been studied on an ad hoc basis by many authors. Sufficient conditions, but usually not necessary conditions, for the stability of these cycles have been obtained via a variety of different techniques.

We begin a systematic investigation into the asymptotic stability of such cycles. A general sufficient condition for asymptotic stability is obtained, together with algebraic criteria for deciding when this condition is also necessary. These criteria are always satisfied in ℝ3 and often satisfied in higher dimensions. We end by applying our results to several higher-dimensional examples that occur in mode interactions with O(2) symmetry.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 15 , Issue 1 , February 1995 , pp. 121 - 147
Copyright
Copyright © Cambridge University Press 1995

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