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 $⟨I^2⟩$ along strongly chaotic orbits, i.e. when $K\gg 1$. We numerically prove that $⟨I^2⟩\propto n^{\mu }$ with $0\le \mu (\alpha )\le 1$, 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 $\alpha \to 1$ 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.

5 pages, 3 figures