On fractional-order maps and their synchronization

TL;DR

This paper investigates the stability and synchronization of linear fractional-order maps, revealing how Mittag-Leffler functions describe evolution and how stability criteria depend on fractional parameters, with extensions to nonlinear maps.

Contribution

It provides a comprehensive analysis of stability conditions for fractional maps, introduces normal mode analysis for coupled systems, and extends results to fractional nonlinear maps.

Findings

Mittag-Leffler functions describe evolution in stable fractional maps

Stability regions depend on fractional order parameter and eigenvalues

Normal mode analysis simplifies coupled fractional map stability analysis

Abstract

We study the stability of linear fractional order maps. We show that in the stable region, the evolution is described by Mittag-Leffler functions and a well defined effective Lyapunov exponent can be obtained in these cases. For one-dimensional systems, this exponent can be related to the corresponding fractional differential equation. A fractional equivalent of map $f(x)=ax$ is stable for $a_c(\alpha)<a<1$ where $\alpha$ is a fractional order parameter and $a_c(\alpha)\approx -\alpha$. For coupled linear fractional maps, we can obtain `normal modes' and reduce the evolution to effectively one-dimensional system. If the eigenvalues are real the stability of the coupled system is dictated by the stability of effectively one-dimensional normal modes. For complex eigenvalues, we obtain a much richer picture. However, in the stable region, the evolution of modulus is dictated by Mittag-Leffler function and the effective Lyapunov exponent is determined by modulus of eigenvalues. We extend these studies to synchronized fixed points of fractional nonlinear maps.