The Logistic-Like Map

Abstract

This chapter is dedicated to discuss some dynamical properties of a generalized version of the logistic map called as logistic-like map. The fixed points and their stability are discussed as well as the convergence to the stationary state at and near at the bifurcations. We show the critical exponents defining the convergence to stationary state for the transcritical and supercritical pitchfork bifurcations are not universal and do depend on the nonlinearity of the mapping. On the other hand the critical exponents obtained for the period doubling bifurcations are universal and are independent on the nonlinearity of the map. We use both phenomenological approach with a set of scaling hypothesis and also an approximation transforming the equation of differences into a differential equation in which the solution gives analytically the critical exponents.