Fractional Standard Map

TL;DR

This paper explores the phase space properties of a fractional standard map derived from a kicked fractional differential equation, revealing diverse attractors, chaotic behavior, and fractal-like structures influenced by system parameters.

Contribution

It introduces and analyzes a fractional standard map with memory, highlighting novel attractors and chaotic dynamics not present in the classical standard map.

Findings

Identification of various attractors including fixed points and periodic trajectories

Observation of chaotic trajectories with fractal-like structures

Discovery of sticky attractors in the chaotic sea

Abstract

Properties of the phase space of the standard map with memory are investigated. This map was obtained from a kicked fractional differential equation. Depending on the value of the parameter of the map and the fractional order of the derivative in the original differential equation this nonlinear dynamical system demonstrates attractors (fixed points, stables periodic trajectories, slow converging and slow diverging trajectories, ballistic trajectories, and fractal-like structures) and/or chaotic trajectories. At least one type of fractal-like sticky attractors in the chaotic sea was observed.