Caputo Standard $\alpha$-Family of Maps: Fractional Difference vs. Fractional

TL;DR

This paper compares fractional differential and fractional difference systems using Caputo Standard α-Families of Maps, highlighting similarities and differences in their dynamical behaviors and memory properties.

Contribution

It introduces a comparative analysis of fractional difference and fractional maps, revealing their similar and distinct dynamical features and memory effects.

Findings

Fractional difference maps exhibit properties similar to fractional maps.

Both types show cascade bifurcations and power-law convergence.

Differences are observed in memory effects and attractor structures.

Abstract

In this paper the author compares behaviors of systems which can be described by fractional differential and fractional difference equations using the fractional and fractional difference Caputo Standard $\alpha$-Families of Maps as examples. The author shows that properties of fractional difference maps (systems with falling factorial-law memory) are similar to the properties of fractional maps (systems with power-law memory). The similarities (types of attractors, power-law convergence of trajectories, existence of cascade of bifurcations and intermittent cascade of bifurcations type trajectories, and dependence of properties on the memory parameter $\alpha$) and differences in properties of falling factorial- and power-law memory maps are investigated.