On homoclinic bifurcation emanating from Takens-Bogdanov points in Hamiltonian systems

Abstract

The paper is devoted to studying the bifurcation of periodic and homoclinic orbits in a 2n-dimensional Hamiltonian system with 1 parameter from a TB-point (Hamiltonian saddle node). In addition to the proof of existence, the paper gives an expansion formula of the bifurcating homoclinic orbits. With the help of center manifold reduction and a blow up transformation, the problem is focused on studying a planar Hamiltonian system, the proof for the perturbed homoclinic and periodic orbits is elementary in the sense that it uses only implicit function arguments. Two applications to travelling waves in PDEs are shown.