Coexisting cycles in a class of 3-D discrete maps

In this paper we consider the class of three-dimensional discrete maps M (x, y, z) =[ Phi(y) ,Phi(z) , Phi(x)], where Phi: R --> R is an endomorphism. We show that all the cycles of the 3-D map M can be obtained by those of Phi(x), as well as their local bifurcations. In particular we obtain that any local bifurcation is of co-dimension 3, that is three eigenvalues cross simultaneously the unit circle. As the map M exhibits coexistence of cycles when Phi(x) has a cycle of period n>1, making use of the Myrberg map as endomorphism, we describe the structure of the basins of attraction of the attractors of M and we study the eff ect of the fl ip bifurcation of a fi xed point.