Fractional Maps and Fractional Attractors. Part II: Fractional Difference $\alpha$-Families of Maps

TL;DR

This paper introduces fractional difference $eta$-families of maps with power-law memory, extending discrete nonlinear systems analysis using Caputo difference operators to explore their properties.

Contribution

It extends the concept of $eta$-families of maps to fractional difference equations with power-law memory, providing new tools for analyzing discrete systems.

Findings

Defined fractional difference $eta$-families of maps.

Linked fractional difference maps to power-law memory behavior.

Proposed new models for discrete nonlinear systems with memory.

Abstract

In this paper we extend the notion of an $\alpha$-family of maps to discrete systems defined by simple difference equations with the fractional Caputo difference operator. The equations considered are equivalent to maps with falling factorial-law memory which is asymptotically power-law memory. We introduce the fractional difference Universal, Standard, and Logistic $\alpha$-Families of Maps and propose to use them to study general properties of discrete nonlinear systems with asymptotically power-law memory.