The Parameter Space Structure of the Kicked Logistic Map and Its Stability

Abstract

In this paper, we analyzed the three-dimension parameter space of the Kicked Logistic Map (KLM), which is the Logistic Map perturbed by periodic kicks with constant amplitude. In this space parameter, diagrams are numerically determined, identifying the regions with finite attractors and their topology. For the identified periodic regions, isoperiodic diagrams are also computed. Examples of these diagrams are presented for fixed kick periods. Dynamical properties of the KLM are characterized by the forms observed in these diagrams. Furthermore, the considered map has different basins of attraction. Thus, for the kick period t = 2, an analytical analysis shows the coexistence of two basins of attraction. In addition, for this kick period, a stability diagram is presented for the period-two orbits, without iterating the KLM, reproducing the corresponding regions in the isoperiodic diagrams. Lastly, the coexistence of two basins, one for a periodic and another for a chaotic attractor, causes, in critical regions of the parameter space, the appearance of a type of crisis named transfer crisis.