Discrete Map with Memory from Fractional Differential Equation of Arbitrary Positive Order

TL;DR

This paper introduces a new class of discrete maps with power-law memory derived from nonlinear fractional differential equations of arbitrary positive order, generalizing the universal map to include long-term memory effects.

Contribution

It develops a novel method to derive discrete maps with power-law memory from fractional differential equations of any positive order, extending existing models.

Findings

Discrete maps incorporate long-term memory effects.

Maps are generalizations of the universal map.

Memory weights follow a power-law distribution.

Abstract

Derivatives of fractional order with respect to time describe long-term memory effects. Using nonlinear differential equation with Caputo fractional derivative of arbitrary order $\alpha>0$, we obtain discrete maps with power-law memory. These maps are generalizations of well-known universal map. The memory in these maps means that their present state is determined by all past states with power-law forms of weights. Discrete map equations are obtained by using the equivalence of the Cauchy-type problem for fractional differential equation and the nonlinear Volterra integral equation of the second kind.