Singular Heteroclinic Cycles

We study the unfolding of heteroclinic cycles of vector fields in n, that possess a hyperbolic singularity and a saddle-node. The principal eigenvalues at the hyperbolic singularity are assumed to be real, but the weak hyperbolic eigenvalues at the saddle-node may be either real or complex conjugate. We discuss the bifurcation diagrams of all codimension two such bifurcations. In some of the occurring cases chaotic dynamics appears in the unfolding. For the cases with only simple dynamics in the unfolding, we obtain the complete bifurcation diagram. We derive exponential expansions for the transition map near the saddle-node.