Long-term memory contribution as applied to the motion of discrete dynamical systems

TL;DR

This paper investigates how long-term memory influences the dynamics of logistic maps, showing that increased memory suppresses chaos and alters bifurcation structures, with the memory effect governed by a parameter .

Contribution

It introduces a model incorporating long-term memory into logistic maps and analyzes how this affects bifurcation and chaos development.

Findings

Memory effects hinder chaos development in logistic maps.

The parameter  controls bifurcation behavior.

For  > 0.15, chaos is suppressed.

Abstract

We consider the evolution of logistic maps under long-term memory. The memory effects are characterized by one parameter \alpha. If it equals to zero, any memory is absent. This leads to the ordinary discrete dynamical systems. For \alpha = 1 the memory becomes full, and each subsequent state of the corresponding discrete system accumulates all past states with the same weight just as the ordinary integral of first order does in the continuous space. The case with 0 < \alpha < 1 has the long-term memory effects. The characteristic features are also observed for the fractional integral depending on time, and the parameter \alpha is equivalent to the order index of fractional integral. We study the evolution of the bifurcation diagram among \alpha = 0 and \alpha = 0.15 . The main result of this work is that the long-term memory effects make difficulties for developing the chaos motion in such logistic maps. The parameter \alpha\ resembles a governing parameter for the bifurcation diagram. For \alpha\ > 0.15 the memory effects win over chaos.