Tunable subdiffusion in the Caputo fractional standard map

Abstract

The Caputo fractional standard map (C-fSM) is a two-dimensional nonlinear map with memory given in action-angle variables $(I,\theta)$. It is parameterized by $K$ and $\alpha\in(1,2]$ which control the strength of nonlinearity and the fractional order of the Caputo derivative, respectively. In this work we perform a scaling study of the average squared action $\left< I^2 \right>$ along strongly chaotic orbits, i.e. when $K\gg1$. We numerically prove that $\left< I^2 \right>\propto n^\mu$ with $0\le\mu(\alpha)\le1$, for large enough discrete times $n$. That is, we demonstrate that the C-fSM displays subdiffusion for $1<\alpha<2$. Specifically, we show that diffusion is suppressed for $\alpha\to1$ since $\mu(1)=0$, while standard diffusion is recovered for $\alpha=2$ where $\mu(2)=1$. We describe our numerical results with a phenomenological analytical estimation. We also contrast the C-fSM with the Riemann-Liouville fSM and Chirikov's standard map.