Scaling properties of the action in the Riemann-Liouville fractional standard map

TL;DR

This study investigates the scaling behavior of the average squared action in the Riemann-Liouville fractional standard map, revealing universal functions and independence from the fractional order in strongly chaotic regimes.

Contribution

The paper provides a detailed scaling analysis of the RL-fSM's action, showing universality and independence from fractional order for large nonlinearity parameters.

Findings

 is a universal function of scaled discrete time $nK^2/I_0^2$

 is independent of  for large K

Analytical results support numerical observations

Abstract

The Riemann-Liouville fractional standard map (RL-fSM) is a two-dimensional nonlinear map with memory given in action-angle variables $(I,\theta)$. The RL-fSM is parameterized by $K$ and $\alpha\in(1,2]$ which control the strength of nonlinearity and the fractional order of the Riemann-Liouville derivative, respectively. In this work, we present a scaling study of the average squared action $\left< I^2 \right>$ of the RL-fSM along strongly chaotic orbits, i.e. for $K\gg1$. We observe two scenarios depending on the initial action $I_0$, $I_0\ll K$ or $I_0\gg K$. However, we can show that $\left< I^2 \right>/I_0^2$ is a universal function of the scaled discrete time $nK^2/I_0^2$ ($n$ being the $n$th iteration of the RL-fSM). In addition, we note that $\left< I^2 \right>$ is independent of $\alpha$ for $K\gg1$. Analytical estimations support our numerical results.