Route to chaos and some properties in the boundary crisis of a generalized logistic mapping

Abstract

A generalization of the logistic map is considered, showing two control parameters α and β that can reproduce different logistic mappings, including the traditional second degree logistic map, cubic, quartic and all other degrees. We introduce a parametric perturbation such that the original logistic map control parameter R changes its value periodically according an additional parameter ω=2∕q. The value of q gives this period. For this system, an analytical expression is obtained for the first bifurcation that starts a period-doubling cascade and, using the Feigenbaum Universality, we found numerically the accumulation point Rc where the cascade finishes giving place to chaos. In the second part of the paper we study the death of this chaotic behavior due to a boundary crisis. At the boundary crisis, orbits can reach a maximum value X=Xmax=1. When it occurs, the trajectory is mapped to a fixed point at X=0. We show that there exist a general recursive formula for initial conditions that lead to X=Xmax.