Degenerate bifurcations of heterodimensional cycles with orbit flip

In this paper, nongeneric bifurcation analysis near heterodimensional cycles with orbit flip is investigated for three-dimensional systems. With the aid of a suitable local coordinate system, the Poincaré map is constructed. By means of the bifurcation equations, the existence, nonexistence, coexistence and uniqueness of homoclinic orbit, periodic orbits and the heterodimensional cycle are studied, the relevant bifurcation surfaces and their existing regions are given. Some known results are extended. An example is given to show the existence of the system which has a heterodimensional cycle with orbit flip.