Homoclinic saddle to saddle-focus transitions in 4D systems

We analyze a saddle to saddle-focus transition on a 4-dimensional homoclinic center manifold where the stable/unstable leading eigenspace is 3-dimensional (called the 3DL bifurcation). The transition is different from the Belyakov bifurcation, where a pair of complex eigenvalues split into two distinct real eigenvalues. Here a pair of complex eigenvalues and a real eigenvalue cross each other transversally, giving rise to a 3-dimensional stable/unstable leading situation. The work is motivated by the observation of this transition in [Meijer and Coombes, 2014] for the tame case (negative saddle quantity). We use near-to-saddle Poincare maps to give a detailed picture of global and local bifurcations occurring close to the critical saddle and the homoclinic connection respectively, in both wild and tame cases. We obtain sets of codimension 1 and 2 bifurcations that asymptotically approach the 3DL bifurcation point which are different in structure than those obtained in the Belyakov case.