Poincaré recurrences and transient chaos in systems with leaks

In order to simulate observational and experimental situations, we consider a leak in the phase space of a chaotic dynamical system. We obtain an expression for the escape rate of the survival probability by applying the theory of transient chaos. This expression improves previous estimates based on the properties of the closed system and explains dependencies on the position and size of the leak and on the initial ensemble. With a subtle choice of the initial ensemble, we obtain an equivalence to the classical problem of Poincaré recurrences in closed systems, which is treated in the same framework. Finally, we show how our results apply to weakly chaotic systems and justify a split of the invariant saddle into hyperbolic and nonhyperbolic components, related, respectively, to the intermediate exponential and asymptotic power-law decays of the survival probability.

Figure 1

(Color online) Illustration of the equivalence between recurrence and escape times.

Figure 2

(Color online) Hénon map (21) with a leak at $(x_c,y_c)=(1.0804,0.0916)$ of radius $\delta =0.1$, depicted as a red circle. (a) Different initial densities: ${\rho }_{\mu }$ of the natural measure on the Hénon attractor, ${\rho }_s=\mathrm{const}$ on $0<x<1$, $-0.2<y<0.2$, and ${\rho }_r$, equivalent to recurrence. The inset shows the nontrivial dependence of the projection ${\rho }_r\left(y\right)$ of ${\rho }_r$ on the $y$ axis. (b) Invariant saddle composed of the points that do not enter $I$ for forward and backward iterations. (c) Unstable manifold of the invariant saddle $\left({\rho }_c\right)$ and (d) stable manifold in gray (orange) of the invariant saddle.

Figure 3

(Color online) (a) Escape time distributions for the Hénon map (21) with a leak of radius $\delta =0.05$ at $(x_c,y_c)=(1.0804,0.0916)$. Results for initial densities ${\rho }_r$, ${\rho }_{\mu }$, and ${\rho }_s$ (shown in Fig. 2) are presented from top to bottom, respectively. The thin solid line corresponds to ${\gamma }^{*}=0.0385$ given by (6) and strongly differs from the actual exponent $\gamma =0.545$. The inset shows the short-time behavior. (b) The same as in (a) multiplied by $\exp \left(\gamma n\right)$. Three different procedures were employed to fit $\gamma$ using $p\left(n\right)$ for $n<n^{*}$ (see Sec. 7) and are indicated by the dashed lines (i)–(iii). In all cases $n^{*}=35$ and (i) corresponds to the use of relation (33), (ii) follows from relations (31, 32) with the numerical value of $⟨n⟩$ for ${\rho }_0={\rho }_r$, and (iii) from relations (31, 32) with the numerical value of $⟨n⟩$ for ${\rho }_0={\rho }_s$.

Figure 4

(Color online) Dependence of the inverse mean escape time $1∕⟨n⟩$ on the radius $\delta$ of the leak for fixed $(x_c,y_c)=(0.015714,0.268892)$. Initial densities equivalent to those shown in Fig. 2 are taken. The straight line corresponds to ${\delta }^{-D_0}$, where $D_0=1.27$ is the fractal dimension of the attractor [53]. The inset shows a magnification of the main graph for large $\delta$.

Figure 5

(Color online) (a) Graphical representation of the logistic map (23) with a leak $I$ at $x_c=0.29427$ of half-width $\delta =0.00716$ so that $\mu \left(I\right)=0.01$. The dashed line corresponds to ${\rho }_{\mu }\left(x\right)$, Eq. (24), and the (blue) histogram to a numerical distribution of the $c$ measure of the chaotic repeller obtained by leaking the system at $I$. The dotted (red) histogram corresponds to ${\rho }_r\left(x\right)$ given by (17) multiplied by $\mu \mathbf{\left(}M\left(I\right)\mathbf{\right)}$. (b) Dependence of the inverse mean escape time, multiplied by ${⟨n⟩}_r$, on the leak position $x_c$ in the logistic map (23) at an interval length such that $\mu \left(I\right)=0.01$. The initial densities used are (see Table ) ${\rho }_r$, equivalent to recurrence, ${\rho }_{\mu }$, natural density Eq. (24), and ${\rho }_s$, chosen uniform in $0<x<1$.

Figure 6

(Color online) (a) Phase space of the standard map (28) with $I=[0.25<x<0.25+\epsilon ,-0.5<y<-0.5+\epsilon ]$ of size $\epsilon =0.15$, indicated by a checkered box. The KAM island (nonhyperbolic part of the saddle) is in the center of the figure. The support of ${\rho }_r$ at $M\left(I\right)$ is marked as a dark (red) region. (b) The unstable manifold of the hyperbolic part of the saddle (black dots), obtained by using the method of [26], choosing an escape time $n_s=80<n_c=307$. The support of ${\rho }_r$ is fully outside the unstable manifold.

Figure 7

(Color online) (a) Escape time distributions in the standard map with $\epsilon =0.15$, as shown in Fig. 6. Results are shown for ${\rho }_0={\rho }_r$ (squares below) and for ${\rho }_0={\rho }_{\mu }$ in $\mid x\mid >0.25$. Lower inset: scaling of $n_c\gamma$ with $\gamma$ obtained by changing $\epsilon$ in $0.03⩽\epsilon ⩽0.4$. Upper inset: log-log plot of the main graph over a long period of time. (b) $p_{e,r}$ multiplied by $\exp \left(\gamma n\right)$, where $\gamma =0.027403$.

Figure 8

The saddle characterizing the dynamics at different times, $n_s$, is given on top of the panels, obtained by the method of Ref. 26. For $n_s<n_c=307$ in (a), the hyperbolic component obtained is clearly disjoint from the region of the KAM island. By increasing $n_s$ as in (b) and (c) the nonhyperbolic component appears with an increasing weight.

Figure 9

(Color online) Distribution of the $m\text{th}$ recurrence times ($m=1,\dots ,4$, from bottom to top) in the Hénon map with parameters as in Fig. 3. The solid lines correspond to the binomial distribution (35) using $\mu ={\mu }_c\left(I\right)$. For long times, the same decay rate $\gamma =\ln [1-{\mu }_c\left(I\right)]$ is observed in all curves. The displacement between the distribution and the simulation, as in the $m=1$ case discussed previously, can be related to the short-time oscillations which modify the prefactors of the distribution.