Bifurcation and Chaos in the Logistic Map with Memory

Abstract

We describe the dynamics of the logistic map [Formula: see text] endowed with memory [Formula: see text] of past iterations with geometric decay. We find that most of the relevant properties of the dynamics can be explained by means of a scaling of the standard logistic map without memory. The bifurcation diagram is therefore a scaled version of the standard map, with the extreme values of the logistic parameter [Formula: see text] for confined trajectories following also the scaling law. However, contrary to the ahistoric model, the trajectories may diverge for smaller values of [Formula: see text], depending on the initial condition [Formula: see text]. Simple conditions for divergence with given logistic map and memory are provided.