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 \lambda of the system is modified. Complex patterns on the plane are visualised as a consequence of basins' bifurcations. According to the already established nomenclature in the literature, we present the relevant phenomenology organised 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. 
Ricardo Lopez-Ruiz and Daniele Fournier-Prunaret