Complex Patterns on the Plane: Different Types of Basin Fractalization in a Two-Dimensional Mapping

Basins generated by a noninvertible mapping formed by two symmetrically coupled logistic maps are studied when the only parameter λ of the system is modified. Complex patterns on the plane are visualized as a consequence of the basins’ bifurcations. According to the already established nomenclature in the literature, we present the relevant phenomenology organized in different scenarios: fractal islands disaggregation, finite disaggregation, infinitely disconnected basin, infinitely many converging sequences of lakes, countable self-similar disaggregation and sharp fractal boundary. By use of critical curves, we determine the influence of zones with different number of first rank preimages in the mechanisms of basin fractalization.