Structurally stable heteroclinic cycles

Source abstract

This paper describes a previously undocumented phenomenon in dynamical systems theory; namely, the occurrence of heteroclinic cycles that are structurally stable within the space of C r vector fields equivariant with respect to a symmetry group. In the space X ( M ) of C r vector fields on a manifold M , there is a residual set of vector fields having no trajectories joining saddle points with stable manifolds of the same dimension. Such heteroclinic connections are a structurally unstable phenomenon [4]. However, in the space X G ( M ) ⊂ X ( M ) of vector fields equivariant with respect to a symmetry group G , the situation can be quite different. We give an example of an open set U of topologically equivalent vector fields in the space of vector fields on ℝ 3 equivariant with respect to a particular finite subgroup G ⊂ O (3) such that each X ∈ U has a heteroclinic cycle that is an attractor. The heteroclinic cycles consist of three equilibrium points and three trajectories joining them.