Codimension-3 Flip Bifurcation Of A Class Of Difference Equations

In this paper, we consider a one-dimensional difference equation with three parameters, the derivative of which at a fixed point has an eigenvalue - 1 as the parameters are all zero. In the case that both nondegeneracy conditions of the flip bifurcation and the generalized flip bifurcation are not satisfied, by computing normal form, we give the nondegeneracy condition and transversality condition of the codimension-3 flip bifurcation. Moreover, by discussing the number of positive zeros of a cubic function in a neighborhood of the origin, we show the bifurcation scenario and give the parameter conditions, respectively, that the normal form of the equation possesses three 2-cycles, two 2-cycles, only one 2-cycle or none.