Proc. Roy. Soc. Edinburgh A 134A (2004) 1177-1197
Martin Krupa and Ian Melbourne
Abstract
Systems possessing symmetries often admit robust heteroclinic cycles that persist under perturbations that respect the symmetry. In previous work, we began a systematic investigation into the asymptotic stability of such cycles. In particular, we found a sufficient condition for asymptotic stability, and we gave algebraic criteria for deciding when this condition is also necessary. These criteria are always satisfied for cycles in R3.
Field & Swift and Hofbauer considered examples in R4 for which our sufficient condition for stability is not optimal. They obtained necessary and sufficient conditions for asymptotic stability using a transition matrix technique.
In this paper, we combine our previous methods with the transition matrix technique and obtain necessary and sufficient conditions for asymptotic stability for a larger class of heteroclinic cycles. In particular, we obtain a complete theory for ``simple'' heteroclinic cycles in R4 (thereby proving and extending results for homoclinic cycles that were stated without proof by Chossat, Krupa, Melbourne and Scheel ). A partial classification of simple heteroclinic cycles in R4 is also given. Finally, our stability results generalise naturally to higher dimensions and many of the higher-dimensional examples in the literature are covered by this theory.