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 $(I,θ)$. It is parameterized by $K$ and $α\in(1,2]$ 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 $\left< I^2 \right>\propto n^μ$ with $0\leμ(α)\le1$, 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 $α\to1$ 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.

Figures (3)

• Figure 1: Average squared action $\left< I^2_n \right>_{\hbox{\tiny int}}$ as a function of $n$ for (a) $(I_0,K)=(10^2,10^4)$, (b) $(I_0,K)=(10^2,10^2)$, and (c) $(I_0,K)=(10^3,10^2)$. Several values of $\alpha$ are considered, as indicated in panel (c). Red-dashed lines correspond to Eq. (\ref{['I2RLfSM']}). Blue-dashed lines are Eq. (\ref{['I2intCfSM']}). The average is taken over $M=200$ orbits with initial random phases in the interval $0<\theta_0<2\pi$.
• Figure 2: (a) Average squared action $\left< I^2_n \right>_{\hbox{\tiny int}}$ as a function of $n$ for $K=10^2$ (blue symbols), $K=10^4$ (red symbols), and $K=10^6$ (black symbols). In all cases $I_0=0$. The average is taken over $M=200$ orbits with initial random phases in the interval $0<\theta_0<2\pi$. Several values of $\alpha$ are considered; same symbol labeling as in Fig. \ref{['Fig02']}. Dashed lines correspond to Eq. (\ref{['I2RLfSM']}). (b) $\left< I^2_n \right>_{\hbox{\tiny int}}/K^2$ vs. $n$. Same data as in panel (a). (c) $\left< I^2_n \right>_{\hbox{\tiny int}}$ vs. $n$ for $K=10^3$ and $I_0=0$. Here the average is taken over $M=100$ orbits with initial random phases in the interval $0<\theta_0<2\pi$. Dashed lines correspond to power-law fittings of the form $\left< I^2_n \right>_{\hbox{\tiny int}}\propto n^\mu$ in the interval $n=[10^4,10^6]$. (d) $\mu$, from the power-law fittings of panel (c), as a function of $\alpha$. The red-dashed line is a linear fit to the data with $\alpha>0.5$: $\mu\sim1.69\alpha$.
• Figure 3: (a) $f(\alpha)$ and $f(\alpha)/[\Gamma(\alpha-1)]^2$. $f(\alpha)$ is obtained from the power-law fittings of the form $\left< I^2_n \right>_{\hbox{\tiny int}}= {\cal C} n^\mu$ to the data of Fig. \ref{['Fig03']}(c); i.e. $f(\alpha)=2{\cal C}[\mu(\alpha) + 1][\Gamma(\alpha-1)]^2/K^2$, here with $K=10^3$. (b) $n^*(\alpha)$ for three ratios $I_0/K$; see Eq. (\ref{['nCO']}).