Universality in Systems with Power-Law Memory and Fractional Dynamics

TL;DR

This paper reviews how introducing power-law memory and fractional derivatives into nonlinear dynamical systems extends classical universality and chaos phenomena, revealing new bifurcation behaviors unique to systems with memory.

Contribution

It presents a comprehensive overview of how fractional dynamics and power-law memory alter the universality and bifurcation cascades in nonlinear maps, highlighting new features in chaotic transitions.

Findings

Fractional maps exhibit cascades of bifurcations on single trajectories.

Universality extends to systems with power-law memory, showing new bifurcation phenomena.

Fractional systems converge to classical maps as the order approaches integers.

Abstract

There are a few different ways to extend regular nonlinear dynamical systems by introducing power-law memory or considering fractional differential/difference equations instead of integer ones. This extension allows the introduction of families of nonlinear dynamical systems converging to regular systems in the case of an integer power-law memory or an integer order of derivatives/differences. The examples considered in this review include the logistic family of maps (converging in the case of the first order difference to the regular logistic map), the universal family of maps, and the standard family of maps (the latter two converging, in the case of the second difference, to the regular universal and standard maps). Correspondingly, the phenomenon of transition to chaos through a period doubling cascade of bifurcations in regular nonlinear systems, known as "universality", can be extended to fractional maps, which are maps with power-/asymptotically power-law memory. The new features of universality, including cascades of bifurcations on single trajectories, which appear in fractional (with memory) nonlinear dynamical systems are the main subject of this review.