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 $\langle I^2\rangle$ 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 $\langle I^2\rangle /I_0^2$ is a universal function of the scaled discrete time $n{K}^2/I_0^2$ ($n$ being the $n\mathrm{th}$ iteration of the RL-fSM). In addition, we note that $\langle I^2\rangle$ is independent of $\alpha$ for $K\gg 1$. Analytical estimations support our numerical results.

Figure 1

Average squared action ${\langle I_n^2\rangle }_{\text{int}}$ as a function of the discrete time $n$ for (a) $\alpha =1.1$, (b) $\alpha =1.5$, and (c) $\alpha =1.9$. Open symbols (full symbols) correspond to $I_0\ll K$ ($I_0\gg K$). The blue dashed lines, plotted to guide the eye, are proportional to $n$. The average is over 100 orbits with initial random phases in the interval $0<{\theta }_0<2\pi$. Red full lines are Eq. (9).

Figure 2

(a), (b) Average squared action ${\langle I_n^2\rangle }_{\text{int}}$ as a function of $n$ for (a) $K={10}^3$ and several values of $I_0$ ($2\times {10}^3$, $4\times {10}^3$, ${10}^4$, $2\times {10}^4$, $4\times {10}^4$, ${10}^5$, and $2\times {10}^5$, from bottom to top) and (b) $I_0={10}^5$ and several values of $K$ ($4\times {10}^2$, ${10}^3$, $2\times {10}^3$, $4\times {10}^3$, ${10}^4$, $2\times {10}^4$, and $4\times {10}^4$, from bottom to top). As in Fig. 1, the average is taken over 100 orbits with initial random phases in the interval $0<{\theta }_0<2\pi$. All data in (a), (b) correspond to $\alpha =1.1$. The horizontal red dashed lines in (a), (b) indicate ${\langle I_n^2\rangle }_{\text{int}}=I_0^2$ with (a) $I_0=4\times {10}^4$ and (b) $I_0={10}^5$, respectively. The transverse red dashed lines in (a), (b) are fittings of ${\langle I_n^2\rangle }_{\text{int}}=\mathcal{C}n$ to the data represented by asterisks (for $n\ge {10}^4$) with fitting constants (a) $\mathcal{C}=294658$ and (b) $\mathcal{C}=3984348$. (c), (d) Crossover time $n^{*}$ (c) as a function of $I_0$ for constant $K$ ($K={10}^3$) and (d) as a function of $K$ for constant $I_0$ ($I_0={10}^5$). In (c), (d), three values of $\alpha$ are reported: $\alpha =1.1$, 1.5, and 1.9. Red dashed lines in (c), (d) are power-law fittings to the data of the form (a) $n^{*}\propto I_0^{{\gamma }_1}$ with ${\gamma }_1\approx 2$ and (b) $n^{*}\propto K^{{\gamma }_2}$ with ${\gamma }_2\approx -2$. Blue dot-dashed lines in (c), (d) are Eq. (11).

Figure 3

Normalized average squared action ${\langle I_n^2\rangle }_{\text{int}}/I_0^2$ as a function of the normalized time $n{K}^2/I_0^2$. Same data sets as in Fig. 1. The red line is Eq. (10).