Differential Equations with Fractional Derivative and Universal Map with Memory

TL;DR

This paper derives discrete maps with long-term memory from fractional differential equations, generalizing the universal map by incorporating past states with specific weights using Riemann-Liouville and Caputo derivatives.

Contribution

It introduces a unified approach to derive universal maps with memory from fractional differential equations considering general initial conditions.

Findings

Derived discrete maps with memory from fractional derivatives.

Unified framework for Riemann-Liouville and Caputo derivatives.

Accounts for general initial conditions in maps with memory.

Abstract

Discrete maps with long-term memory are obtained from nonlinear differential equations with Riemann-Liouville and Caputo fractional derivatives. These maps are generalizations of the well-known universal map. The memory means that their present state is determined by all past states with special forms of weights. To obtain discrete map from fractional differential equations, we use the equivalence of the Cauchy-type problems and to the nonlinear Volterra integral equations of second kind. General forms of the universal maps with memory, which take into account general initial conditions, for the cases of the Riemann-Liouville and Caputo fractional derivatives, are suggested.