Bifurcations Near the Weak Type Heterodimensional Cycle

Abstract

The aim of this paper is to show the bifurcation phenomena near the weak type heterodimensional cycle when the orbit flip and inclination flip occur simultaneously in its nontransversal heteroclinic orbit. With the aid of a suitable local coordinate system, the Poincaré map is constructed. By means of the bifurcation equations, the persistence of heterodimensional cycles, the coexistence of the heterodimensional cycle and periodic orbits or homoclinic orbits, and the existence of bifurcation surfaces of homoclinic orbits or the periodic orbits are discussed under small perturbations. Moreover, an example is given to show the existence of the system which has a heterodimensional cycle with orbit flip and inclination flip.