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

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 $<I^2>$ of the RL-fSM along strongly chaotic orbits, i.e. for $K\gg 1$. 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 $<I^2>\mathrm{/}{I}_0^2$ is a universal function of the scaled discrete time $n{K}^2\mathrm{/}{I}_0^2$ ($n$ being the $n$th iteration of the RL-fSM). In addition, we note that $<I^2>$ is independent of $\alpha$ for $K\gg 1$. Analytical estimations support our numerical results.