Dissipative fractional standard maps: Riemann-Liouville and Caputo

TL;DR

This paper investigates how dissipation affects fractional standard maps with memory, revealing exponential decay of action and the influence of fractional order on squared action, with analytical expressions derived for the Riemann-Liouville case.

Contribution

It introduces dissipative fractional standard maps with Riemann-Liouville and Caputo derivatives, analyzing their dynamics and deriving analytical expressions for key quantities.

Findings

Dissipation causes exponential decay of average action in both maps.

The Caputo map's squared action is strongly affected by fractional order when alpha<2.

An analytical expression for the squared action in the Riemann-Liouville map is provided.

Abstract

In this study, given the inherent nature of dissipation in realistic dynamical systems, we explore the effects of dissipation within the context of fractional dynamics. Specifically, we consider the dissipative versions of two well known fractional maps: the Riemann-Liouville (RL) and the Caputo (C) fractional standard maps (fSMs). Both fSMs are two-dimensional nonlinear maps with memory given in action-angle variables $(I_n,\theta_n)$; $n$ being the discrete iteration time of the maps. In the dissipative versions these fSMs are parameterized by the strength of nonlinearity $K$, the fractional order of the derivative $\alpha\in(1,2]$, and the dissipation strength $\gamma\in(0,1]$. In this work we focus on the average action $\left< I_n \right>$ and the average squared action $\left< I_n^2 \right>$ when~$K\gg1$, i.e. along strongly chaotic orbits. We first demonstrate, for $|I_0|>K$, that dissipation produces the exponential decay of the average action $\left< I_n \right> \approx I_0\exp(-\gamma n)$ in both dissipative fSMs. Then, we show that while $\left< I_n^2 \right>_{RL-fSM}$ barely depends on $\alpha$ (effects are visible only when $\alpha\to 1$), any $\alpha< 2$ strongly influences the behavior of $\left< I_n^2 \right>_{C-fSM}$. We also derive an analytical expression able to describe $\left< I_n^2 \right>_{RL-fSM}(K,\alpha,\gamma)$.