Title:

Heterodimensional cycles near homoclinic bifurcations

In this thesis we study bifurcations of a pair of homoclinic loops to a saddlefocus equilibrium (with a onedimensional unstable manifold) in flows with dimension four or higher. Particularly, we show that heterodimensional cycles can be born from such bifurcations. A heterodimensional cycle consists of two saddle periodic orbits having different indices (dimensions of unstable manifolds), and two heteroclinic connections between those orbits. We find heterodimensional cycles for the flow as the suspension of heterodimensional cycles for a Poincaré map around the homoclinic loops. Especially, those cycles are coindex 1, i.e. the difference between indices is 1. More specifically, each of those heterodimensional cycles are associated to periodic orbits of indices 2 and 3. As a partial result we mention a criterion for having index 3 for periodic orbits near a homoclinic loop to a saddlefocus equilibrium. Different types of perturbations are considered, where the original homoclinic loops can be either kept or split. In intermediate steps we find, in addition to the classical heterodimensional connection between two periodic orbits, two new types of heterodimensional connections: one is a heteroclinic between a homoclinic loop and a periodic orbit of index 2, and the other connects a saddlefocus equilibrium to a periodic orbit of index 3. Furthermore, we consider a symmetric case where the codimension of the bifurcations is minimised to 1. We prove that, by endowing the flow with a certain $\mathbb{Z}_2$ symmetry, a pair of heterodimensional cycles can be born from a oneparameter unfolding of the symmetric pair of homoclinic loops. Moreover, we show that the heterodimensional cycles obtained in either the general or the symmetric case can belong to a chaintransitive and volumehyperbolic attractor of the flow, along with a persistent homoclinic tangency.
