Almost Planar Homoclinic Loops in 3

In this paper we study homoclinic loops of vector fields in 3-dimensional space when the two principal eigenvalues are real of opposite sign, which we call almost planar. We are interested to have a theory for higher codimension bifurcations. Almost planar homoclinic loop bifurcations generically occur in two versions “non-twisted” and “twisted” loops. We consider high codimension homoclinic loop bifurcations under generic conditions. The generic condition forces the existence of a 2-dimensional topological invariant ring (non necessarily unique), which is a topological cylinder in the “non-twisted” case and a topological Möbius band in the “twisted” case. If the third eigenvalue is negative (resp. positive) theω-limit set (resp.α-limit set) of trajectories starting in a neighborhood of the loop is contained in the invariant ring. The dynamics is then given by fixed points or periodic points of period 2 of a 1-dimensional map which admits a nice asymptotic expansion, allowing to treat homoclinic loop bifurcations of arbitrarily high codimension in the non-twisted case.