Ergodic Theory and Dynamical Systems

Research Article

Asymptotic stability of heteroclinic cycles in systems with symmetry

Martin Krupaa1 and Ian Melbournea2

a1 Department of Mathematics, University of Groningen, PO Box 800, 9700 AV Groningen, The Netherlands

a2 Department of Mathematics, University of Houston, Houston, Texas 77204-3476, USA

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 3 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.

(Received February 02 1993)

(Revised February 11 1994)