Anomalous dynamics and the choice of Poincaré recurrence set

We investigate the dependence of Poincaré recurrence-time statistics on the choice of recurrence set by sampling the dynamics of two- and four-dimensional Hamiltonian maps. We derive a method that allows us to visualize the direct relation between the shape of a recurrence set and the values of its return probability distribution in arbitrary phase-space dimensions. Such a procedure, which is shown to be quite effective in the detection of tiny regions of regular motion, allows us to explain why similar recurrence sets have very different distributions and how to modify them in order to enhance their return probabilities. Applied to data, this enables us to understand the coexistence of extremely long, transient powerlike decays whose anomalous exponent depends on the chosen recurrence set.

Figure 1

Numerical orbit densities in the phase space $(x_1,y_1)\in {[0,1]}^2$ of map (1) with $K_1=0.65$ and $K_2=0.60$. Panel (a) is the uncoupled case $\beta =0$ (single standard map) after $T={10}^7$ iterations; panels (b), (c), and (d) are magnifications of the square ${[0.2,0.8]}^2$ [depicted in panel (a)] for $\beta ={10}^{-3}$, respectively, after $T={10}^7, {10}^8$, and ${10}^9$ iterations. While the standard-map density exhibits accumulations [panel (a), purple (dark gray) cells] at the borders of stability islands (white indicates unvisited cells), the latter become accessible to the coupled map, although at extremely slow rates: at equal times [panel (b)], the system just started to penetrate areas forbidden to the standard map. Only at much longer times $\sim {10}^9$ [panel (d)] does the phase-space exploration appear complete, but anisotropies are still present. Such filling time scale diverges in the limit of vanishing coupling $\beta \to 0$, while there is a threshold ${\beta }_{*}<\beta$ above which map (1) is completely ergodic [26, 27] (see Sec. 4c).

Figure 2

(a) Scheme of a single Poincaré cycle as described in Sec. 3 for one uncoupled ($\beta =0$) standard map in (1) and recurrence set $B$ [bordered by a pink (gray) dashed line] in the phase space $\mathrm{\Omega }={[0,1]}^2$ (largest square), with departure set $D_B=B_1\setminus B$ [green (gray) zone] and arrival set $A_B=B\setminus B_1$ [yellow (light gray) zone]; recurrence (residence) cycles take place only in the respective sets $RE{C}_B/RE{S}_B$. In these coordinates, the first iterated set $B_1=\mathbf{f}\left(B\right)$ [bordered by a blue (black) dash-dotted line] has the same shape of $B$ reflected about the bisector. (b) Picture of the four sets $D_B, RE{C}_B, A_B$, and $RE{S}_B$ as described in Appendix pp2 for a dynamics $B\to B_1=\mathbf{f}\left(B\right)$ induced by evolving a generic flow $\mathbf{f}={\mathcal{F}}^{\mathrm{\Delta }t}$ along a fixed time step $\mathrm{\Delta }t$; the standard map in panel (a) belongs to the special class of time-periodic Hamiltonian flows whose integration after one period $\mathrm{\Delta }t$ admits an explicit expression, as the one used in Eq. (1).

Figure 3

(a) Poincaré-time (PT) function for the same uncoupled standard map as in Fig. 1 with recurrence set $B_{\left[0.1\right]}=\left\{\right|x_1|<0.1\}$ (gray rectangles), departure set $D_B$ [green (dark gray) rectangles], and PT level sets $S_B^{\tau }$ as in Eq. (D2) with colors encoded by the $\tau$ axis in Fig. 4; white color indicates cells unvisited by Poincaré cycles. (b) Magnification of the square $[0.39,0.41]\times [0.71,0.73]$ from panel (a), detecting islands structures of extremely small area $\sim {10}^{-8}$. (c) and (d) Same PT analysis as panel (a) for two lag sets $I_B^{\delta \tau }$ as in Eq. (5) (gray cells), respectively, for $\delta \tau =5$ and 10. The lag-set departing volume $\mu \left(D_{I_B^{\delta \tau }}\right)$ [green (dark gray) cells indicated by arrows] shrinks as we increase $\delta \tau$.

Figure 4

RT probability (light gray continuous curve) for the recurrence set $B$ in Fig. 3, panel (a) and its lag sets $I_B^{\delta \tau }$ with lags $\delta \tau =5$ [dashed red (black) curve, set in Fig. 3] and $\delta \tau =10$ [dotted red (black) curve, set in Fig. 3] as in Eq. (6). The predicted probabilities for $I^{\delta \tau }\left(B\right)$ are matched in the case $\delta \tau =5$ [continuous gray curve for the set $I^5\left(B_{\left[0.1\right]}\right)$ in Fig. 3] while they are underestimated in the case $\delta \tau =10$ [dash-dotted dark gray curve for the set $I^{10}\left(B_{\left[0.1\right]}\right)$ in Fig. 3]. Despite this, both cases produce RT probability enhancement.

Figure 5

Same PT function analysis as in Fig. 3 in phase space $(x_1,y_1)={[0,1]}^2$ for the first of the two maps in Eq. (1) at four coupling regimes: $\beta ={10}^{-6}, {10}^{-5}, {10}^{-4}$, and ${10}^{-3}$ (from the first to the fourth row, respectively) for the recurrence sets $B_{\left[0.1\right]}=\left\{\right|x_1|<0.1\}$ and $B_{\left[0.3\right]}$ and their lag sets $I^{\delta \tau }\left(B_{\left[0.1\right]}\right)$ and $I^{\delta \tau }\left(B_{\left[0.3\right]}\right)$ (gray zones, from the first to the fourth column, respectively) with lag time $\delta \tau =200$ in Eq. (5). Colors are encoded by the $\tau$ axis in Fig. 6, as $\mu \left(S_B^{\tau }\right)\propto P_B\left(\tau \right)$ by Eq. (4), with green (dark gray) for $D_B$ (for lag sets $I^{\delta \tau }$, the latter is made of a few cells) and white for unvisited cells.

Figure 6

RT probability for recurrence sets $B_{\left[0.1\right]}$ and $B_{\left[0.3\right]}$ [gray curves in panels (a) and (b), respectively] pictured in Fig. 3 in the first and third column, respectively. The line types correspond to $\beta ={10}^{-3}$ (dashed line), ${10}^{-4}$ (continuous line), ${10}^{-5}$ (dotted line), and ${10}^{-6}$ (dash-dotted line). The two families of gray curves correspond to $I^{\delta \tau }\left(B_{\left[0.1\right]}\right)$ [panel (a)] and $I^{\delta \tau }\left(B_{\left[0.3\right]}\right)$ [panel (b)] with lag $\delta \tau =200$ and pictured in Fig. 3 in the second and fourth column, respectively. Thin red (dark gray) dotted curves in both panels are the light gray continuous ones ($\beta ={10}^{-4}$) shifted by $\delta \tau$ as in Eq. (6) for the exact lag sets.