Bifurcations Of Heteroclinic Loop With Orbit Flip

In this paper we study the bifurcations of degenerate heterodimensional cycles,which connect-ing one hyperbolic saddle fixed point and one nonhyperbolic fixed point.Also the model of the bi-furcations we study in this paper accompanied by orbit flip and equilibrium bifurcation.First,based on the normal forms theory and the initial conditions in a small neighbourhood of the fixed points we obtain two local mappings,we also obtain the other two regular mappings by taking the first-order term in the Taylor expansion in a small neighbourhood of the heteroclinic orbits.we combine the four mappings and get the Poincare mapping so as to obtain the bifurcation equations.and then we obtain the conditions for the existence of the heteroclinic orbit, homoclinic orbit and periodic orbit by bifurcation equations.Last we analysis the saddle-node bifurcation of the fixed points in a samll neighbourhood of the perturbed heterodimensional cycles with the floquet multipliers theory.