On nonlinear fractional maps: Nonlinear maps with power-law memory

TL;DR

This paper reviews recent findings on nonlinear fractional maps with power-law memory, highlighting their unique attractors, bifurcation behaviors, and convergence properties as they relate to fractional differential equations.

Contribution

It provides a concise overview of the novel dynamical phenomena exhibited by nonlinear fractional maps with power-law memory.

Findings

Discovery of cascade of bifurcations type trajectories

Power-law convergence and divergence of trajectories

Intersection and overlapping of attractors

Abstract

This article is a short review of the recent results on properties of nonlinear fractional maps which are maps with power- or asymptotically power-law memory. These maps demonstrate the new type of attractors - cascade of bifurcations type trajectories, power-law convergence/divergence of trajectories, period doubling bifurcations with changes in the memory parameter, intersection of trajectories, and overlapping of attractors. In the limit of small time steps these maps converge to nonlinear fractional differential equations.