Basin fractalization due to focal points in a class of triangular maps

For a class of rational triangular maps of a plane, characterized by the presence of points in which a component assumes the form 0/0 a new type of bifurcation is evidenced which creates loops in the boundaries of the basins of attraction. In order to explain such bifurcation mechanism, new concepts of focal point and line of focal values are defined, and their effects on the geometric behavior of the map and of its inverses are studied in detail. We prove that the creation of loops, which generally constitute the boundaries of lobes of the basins issuing from the focal points, is determined by contacts between basin boundaries and the line of focal values. A particular map is proposed for which the sequence of such contact bifurcations occurs, causing a fractalization of basin boundaries. Through the analytical and the numerical study of this example new structures of the basins of attraction are evidenced, characterized by fans of stable sets issuing from the focal points, assuming the shape of lobes and arcs, the latter created by the merging of lobes due to contacts between the basin boundaries and the critical curve LC.