Heteroclinic Cycles in Symmetrically Coupled Systems
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 ), it only gained wide attention in the dynamics community after the work of Guckenheimer and Holmes  on a dynamical system used by Busse and Clever  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.
KeywordsWreath Product Internal Symmetry Heteroclinic Cycle Transitive Subgroup Hyperbolic Equilibrium
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