Asymptotic Stability of Heteroclinic Cycles in Systems with Symmetry

Ergodic Theory Dyn. Syst. 15 (1995) 121-147

Martin Krupa and Ian Melbourne


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 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.

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