Structurally stable heteroclinic cycles

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 vector fields equivariant with respect to a symmetry group. In the space X(M) of C 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 XG(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 equivariant with respect to a particular finite subgroup G a 0(3) such that each XeU has a heteroclinic cycle that is an attractor. The heteroclinic cycles consist of three equilibrium points and three trajectories joining them. The system we describe was first discussed by Busse and Heikes[3] in the context of Rayleigh-Benard convection. A similar, but more complicated, phenomenon [1] has been observed recently in models of flow in turbulent boundary layers [2] and by Kevrekides and Nicolaenko [5] in studies of the ' Kolmogorov-Sivashinsky' equation t + xxxx + \x\ 2 + dx((2-8\x\ )x) + fi = 0. Here the group in question is 0(2) acting on R and families of heteroclinic cycles connecting pairs of equilibria are found. The group G a 0(3) that forms the symmetry group of our examples has 24 elements and is generated by