Degenerate bifurcations of heterodimensional cycle with orbit flip and inclination flip

Abstract

We study the heterodimensional cycle with orbit flip and inclination flip in four-dimensional system. By setting up a new local coordinate system in small tubular neighborhood of unperturbed heterodimensional cycle, we construct a Poincare return map and further obtain the bifurcation equations. Based on the bifurcation analysis, we obtain the condition for the existence and coexistence of the heterodimensional orbit, homoclinic orbit and periodic orbit. Also, the relevant bifurcation surfaces and their existing regions are given. Some known results are extended.