Non-resonance 3D homoclinic bifurcation with an inclination flip

Local active coordinates approach is employed to study the bifurcation of a non-resonance three-dimensional smooth system which has a homoclinic orbit to a hyperbolic equilibrium point with three real eigenvalues -α,-β,1 satisfying α>β>0. A homoclinic orbit is called an inclination-flip homoclinic orbit if the strong inclination property of the stable manifold is violated. In this paper, we show the existence of 1-homoclinic orbit, 1-periodic orbit, 2n-homoclinic orbit and 2n-periodic orbit in the unfolding of an inclination-flip homoclinic orbit. And we figure out the bifurcation diagram based on the existence region of the corresponding bifurcation.