Heterodimensional cycle bifurcation with two orbit flips

In this paper, we consider a heteroclinic cycle consisting of two hyperbolic equilibria with different indices, one robust heteroclinic connection and a heteroclinic connection within a codimension-2 intersection of the corresponding manifolds of the equilibria, which is called the heterodimensional cycle. By setting up local moving frame systems in some tubular neighborhood of unperturbed heterodimensional cycles, we construct a Poincaré return map under the nongeneric conditions two orbit flips and further obtain the bifurcation equations. By the bifurcation equations, different bifurcation phenomena are discussed under small perturbations. New features produced by the degeneracy that heterodimensional cycles and periodic orbits coexist on the same bifurcation surface are shown. Some known results are extended. An example is given to show the existence of the system which has a heterodimensional with two orbit flips.