Recurrence-time statistics in non-Hamiltonian volume-preserving maps and flows

We analyze the recurrence-time statistics (RTS) in three-dimensional non-Hamiltonian volume-preserving systems (VPS): an extended standard map and a fluid model. The extended map is a standard map weakly coupled to an extra dimension which contains a deterministic regular, mixed (regular and chaotic), or chaotic motion. The extra dimension strongly enhances the trapping times inducing plateaus and distinct algebraic and exponential decays in the RTS plots. The combined analysis of the RTS with the classification of ordered and chaotic regimes and scaling properties allows us to describe the intricate way trajectories penetrate the previously impenetrable regular islands from the uncoupled case. Essentially the plateaus found in the RTS are related to trajectories that stay for long times inside trapping tubes, not allowing recurrences, and then penetrate diffusively the islands (from the uncoupled case) by a diffusive motion along such tubes in the extra dimension. All asymptotic exponential decays for the RTS are related to an ordered regime (quasiregular motion), and a mixing dynamics is conjectured for the model. These results are compared to the RTS of the standard map with dissipation or noise, showing the peculiarities obtained by using three-dimensional VPS. We also analyze the RTS for a fluid model and show remarkable similarities to the RTS in the extended standard map problem.

Figure 1

Phase-space dynamics of the uncoupled ($\delta =0$) standard map using (a) $K_1=0.09$ (regular dynamics), (b) $K_1=2.6$ (mixed dynamics), (c) $K_1=4.9$ (mixed dynamics), and (d) $K_1=8.8$ (chaotic dynamics). These values will define the dynamics of the extra dimension coupled to the standard map (see Sec. 4). Here we used 80 randomly distributed ICs and $3\times {10}^4$ iterations for each IC.

Figure 2

Illustration of the method used to define the regimes $\mathbf{O}$ and $\mathbf{C}$. The time series of the FTLE ${\lambda }_1^{(\mathrm{\Omega }=100)}$ for the ESM is plotted. The threshold $\varepsilon =0.05$ that defines the regimes is plotted by a dashed-dotted line. A typical initial condition used to generate this time series was chosen inside the chaotic sea.

Figure 3

In top and bottom lines are plotted the cumulative probability distributions $P_{\mathrm{cum}}\left(\tau \right)$ [Eq. (2)] for recurrence times $\tau$ and its scaled version $P_{\mathrm{cum}}\left({\tau }_{\delta }\right)$ [Eq. (4)] respectively, for different values of the coupling $\delta$. For $\delta =0$ we recover results for the standard map. For clarity, the insets in the top line show the distinct behaviors (${\mathbf{O}}_{\mathbf{1}},{\mathbf{O}}_{\mathbf{2}},{\mathbf{O}}_{\mathbf{3}},...$) of the RTS in the ordered regime for $\delta ={10}^{-5}$. In the middle line we also plot $P_{\mathrm{cum}}\left(\tau \right)$ for recurrence times $\tau$ (black continuous lines) together with $P_{\mathrm{cum}}(\tau ={\tau }_M\mathrm{\Omega })$ [Eq. (3)] for the regimes $\mathbf{C}$ (red dashed lines) and $\mathbf{O}$ (blue dashed-dotted lines) for $\delta ={10}^{-5}$. Different values of $K_2$ are used, namely, $K_2=0.09$ (regular dynamics) in (a), (e), and (i); $K_2=2.6$ (mixed dynamics) in (b), (f), and (j); $K_2=4.9$ (mixed dynamics) in (c), (g), and (k) and $K_2=8.8$ (chaotic dynamics) in (d), (h), and (l). In all simulations we used $\mathrm{\Omega }=100$.

Figure 4

Time interval $\mathrm{\Delta }\tau ={\tau }_{p_{\delta }^{\prime }}-{\tau }_{p_{\delta }}$ where the plateaus exist has a function of the inverse of the coupling strength. These times were obtained from the recurrences curves of Figs. 3–3.

Figure 5

Phase space projected in ($x,p$) for $\delta ={10}^{-5}$ and (a) $K_2=0.09$ (regular dynamics), (b) $K_2=2.6$ (mixed dynamics), (c) $K_2=4.9$ (mixed dynamics), and (d) $K_2=8.8$ (chaotic dynamics). The insets show the RTS, the same insets from the top line of Fig. 3, but now with colors indicating the different time intervals ${\mathbf{O}}_{\mathbf{1}},{\mathbf{O}}_{\mathbf{2}},{\mathbf{O}}_{\mathbf{3}},{\mathbf{O}}_{\mathbf{3}}^{\prime },{\mathbf{O}}_{\mathbf{4}}$, and ${\mathbf{O}}_{\mathbf{5}}$, for which the corresponding phase-space points were plotted with the same color.

Figure 6

Three-dimensional phase space ($x,p,w$) for (a) $K_2=0.09$ (regular dynamics); (b) $K_2=2.6$ (mixed dynamics); (c) $K_2=4.9$ (mixed dynamics); and (d) $K_2=8.8$ (chaotic dynamics). These plots correspond, respectively, to Figs. 5–5. Colors denote the same time intervals from Fig. 5. For clarity the times ${\mathbf{O}}_{\mathbf{1}}$ (blue points) from Fig. 5 were not plotted.

Figure 7

Cumulative probability distributions for recurrence times $\tau$ for distinct values of (a) dissipation $\gamma$ and (b) noise intensity $D$.

Figure 8

Cumulative probability distributions for recurrence times $\tau$ for the flow described by Eq. (8) using $b=0.02, \omega =4.0$ and different values of coupling parameter $\epsilon$.