Fractional Standard Map: Riemann-Liouville vs. Caputo

TL;DR

This paper compares the phase space properties of fractional standard maps derived from Riemann-Liouville and Caputo derivatives, revealing unique attractor behaviors and convergence properties due to memory effects.

Contribution

It introduces a detailed analysis of fractional standard maps with memory, highlighting differences between Riemann-Liouville and Caputo derivatives and identifying a new attractor type.

Findings

Attractors in fractional maps differ from classical systems, existing asymptotically.

Trajectories can intersect and lead to the same attractor, unlike in classical dynamics.

A new attractor type, 'cascade of bifurcation trajectories', is common to both maps.

Abstract

Properties of the phase space of the standard maps with memory obtained from the differential equations with the Riemann-Liouville and Caputo derivatives are considered. Properties of the attractors which these fractional dynamical systems demonstrate are different from properties of the regular and chaotic attractors of systems without memory: they exist in the asymptotic sense, different types of trajectories may lead to the same attracting points, trajectories may intersect, and chaotic attractors may overlap. Two maps have significant differences in the types of attractors they demonstrate and convergence of trajectories to the attracting points and trajectories. Still existence of the the most remarkable new type of attractors, "cascade of bifurcation type trajectories", is a common feature of both maps.