Basin Fractalizations Generated by a Two-Dimensional Family of (Z1-z3-z1) Maps

Two-dimensional (Z1–Z3–Z1) maps are such that the plane is divided into three unbounded open regions: a region Z3, whose points generate three real rank-one preimages, bordered by two regions Z1, whose points generate only one real rank-one preimage. This paper is essentially devoted to the study of the structures, and the global bifurcations, of the basins of attraction generated by such maps. In particular, the cases of fractal structure of such basins are considered. For the class of maps considered in this paper, a large variety of dynamic situations is shown, and the bifurcations leading to their occurrence are explained.