N-homoclinic bifurcations for homoclinic orbits changing their twisting

We study bifurcations, called N-homoclinic bifurcations, which produce homoclinic orbits rounding N times (N≥2) in some tubular neighborhood of original homoclinic orbit A family of vector fields undergoes such a bifurcation when it is a perturbation of a vector field with a homoclinic orbit. N-Homoclinic bifurcations are divided into two cases; one is that the linearization at the equilibrium has only real principal eigenvalues, and the other is that it has complex principal eigenvalues. We treat the former case, espcially that linearization has only one unstable eigenvalue. As main tools we use a topological method, namely, Conley index theory, which enables us to treat more degenerate cases than those studied by analytical methods.