Tunable subdiffusion in the Caputo fractional standard map

The Caputo fractional standard map (C-fSM) is a two-dimensional nonlinear map with memory given in action-angle variables (I,θ). It is parameterized by K and α∈(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 < I² > along strongly chaotic orbits, i.e. when K1. We numerically prove that < I² >∝ n^μ with 0≤μ(α)≤1, for large enough discrete times n. That is, we demonstrate that the C-fSM displays subdiffusion for 1<α<2. Specifically, we show that diffusion is suppressed for α→1 since μ(1)=0, while standard diffusion is recovered for α=2 where μ(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.