1999 | OriginalPaper | Chapter
Heteroclinic Cycles in Symmetrically Coupled Systems
Author : Michael Field
Published in: Pattern Formation in Continuous and Coupled Systems
Publisher: Springer New York
Included in: Professional Book Archive
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A characteristic feature of symmetric dynamics is the presence of robust heteroclinic cycles. Although this phenomenon was first described by dos Reis in 1978 (see [25]), it only gained wide attention in the dynamics community after the work of Guckenheimer and Holmes [18] on a dynamical system used by Busse and Clever [7] as a model of fluid convection. Robust heteroclinic cycles also occur in models of population dynamics, see [20, 21]. The existence of robust cycles in equivariant dynamics can be viewed as a special instance of the fact that in equivariant dynamical systems intersections of invariant manifolds can be stable under perturbation even though intersections are not transverse [10, 12]. Indeed, this failure of transversality is a prerequisite for a cycle between hyperbolic equilibria since transversality between stable and unstable manifolds of hyperbolic equilibria implies no cycles.