The existence of infinitely many homoclinic doubling bifurcations from some codimension 3 homoclinic orbits

An inclination-flip homoclinic orbit of weak type on ℝ3 is a homoclinic orbit given as the intersection of a special one-dimensional C2-weak stable manifold and the one-dimensional unstable manifold of a hyperbolic singularity with three real eigenvalues. In this paper, we show that in a generic unfolding of such a homoclinic orbit, there appear curves in the parameter space that correspond to ordinary inclination-flip homoclinic orbit of order N for any integer N. As a consequence, there exist infinitely many homoclinic doubling bifurcation curves emanating from the codimension three degenerate point corresponding to the inclination flip homoclinic orbit of weak type. © 1997 Plenum Publishing Corporation.