Bifurcations Of Heterodimensional Cycle With Orbit Flip

In this paper, we mainly study the problem of a type of heterodimensional cycle with orbit flip in three-dimensional dynamical systems. By setting up a local coordinate around the vicinity of heterodimensional cycles, we give the standard forms of the dynamical system in sufficiently small neighborhood of the two equilib-ria. According to the non-transversal characteristics of heterodimensional cycles, we construct a Poincare return map. Based on the bifurcation analysis of Poincare return map, we present the parametric representation of the non-transversal het-erodimensional cycle, and then obtained the parametric curves of the homoclinic orbits ahd heteroclinic orbits parametric expressions near each of the equilibrium in the parametric plane. Through the deep research on the parametric equation, further more, we establish the images of the parametric curves of the homoclinic orbits and heteroclinic orbits in the parametric plane, and using numerical simu-lation to validate the conclusions. Finally, the expressions of parametric curves of the periodic orbits as well as the approximate shapes of the parametric curves are derived by Poincare return map.