Stochastic perturbations in open chaotic systems: Random versus noisy maps

We investigate the effects of random perturbations on fully chaotic open systems. Perturbations can be applied to each trajectory independently (white noise) or simultaneously to all trajectories (random map). We compare these two scenarios by generalizing the theory of open chaotic systems and introducing a time-dependent conditionally-map-invariant measure. For the same perturbation strength we show that the escape rate of the random map is always larger than that of the noisy map. In random maps we show that the escape rate $\kappa$ and dimensions $D$ of the relevant fractal sets often depend nonmonotonically on the intensity of the random perturbation. We discuss the accuracy (bias) and precision (variance) of finite-size estimators of $\kappa$ and $D$, and show that the improvement of the precision of the estimations with the number of trajectories $N$ is extremely slow ($\propto 1/\ln N$). We also argue that the finite-size $D$ estimators are typically biased. General theoretical results are combined with analytical calculations and numerical simulations in area-preserving baker maps.

Figure 1

Difference between the escape rate in the noisy- ($\tilde{\kappa }$) and random maps ($\widehat{\kappa }$). (a) Number $N\left(t\right)$ of surviving trajectories out of $N\left(0\right)={10}^6$ over time for perturbation strength $\delta =0.1$ in the area-preserving closed random baker map with a central vertical leak of width ${\Delta }_x=0.1$. A number of $R={10}^4$ experiments (realizations of $\xi$) were carried out and (200 of them) are shown as gray lines in the backdrop. The lines in the front show: ${\langle N\rangle }_{r=1}^R$ (blue solid line), corresponding to the noisy map as in Eq. (8); $\exp {\langle \ln N\rangle }_{r=1}^R$ (red dashed line), corresponding to the random map as in Eq. (7); and $\exp ({\langle \ln N\rangle }_{r=1}^R\pm {\sigma }_{\ln N}/2)$ (dash-dotted lines), corresponding to the standard deviation around the expected value in the random-map case. The lower inset shows the histogram of $N(t=15)$ and the upper inset shows the scaling of the dispersion ${\sigma }_{\ln N}$ (half the vertical space between the dash-dotted lines) as $t^{1/2}$ (the solid straight line indicates a slope of 1/2). (b) Dependence of $\tilde{\kappa }$ and $\widehat{\kappa }$ on the perturbation strength $\delta$. A number of $R={10}^4\delta$ experiments are done for each $\delta >0$ [at this time for the noisy map all $N\left(0\right)={10}^6$ trajectories are perturbed independently in all $R$ experiments]. The escape rates $\tilde{\kappa }$ and $\widehat{\kappa }$ were estimated from data from time $t=6$ up to 20 (giving $T=14$). In the case of the random-map error bars indicate the standard deviation ${\sigma }_{\langle \widehat{\kappa }\rangle }$. In the case of the noisy map the error bars ${\sigma }_{\langle \tilde{\kappa }\rangle }$ are smaller than the square marker. In (a) and (b) data points are connected by lines to guide the eye.

Figure 2

Fractal dimension in the area-preserving naturally open baker map. (a) Fractal dimension ${\widehat{D}}_0^{(2,T)}$ vs minimal box size ${\varepsilon }_{*}$. A number of $M={10}^4$ experiments (realizations of $a_t$) were carried out to generate statistics. The standard deviation of ${\widehat{D}}_0^{(2,T)}$ with $T$ fixed is marked by a pair of dashed lines. The lower inset shows the histogram of ${\widehat{D}}_0^{(2,T=500)}$ and the upper inset shows the scaling of the dispersion ${\sigma }_{{\widehat{D}}_0}$ (half the vertical space between the dash-dotted lines, approximately) as ${(\ln 1/{\varepsilon }_{*})}^{-1/2}$ for large values of $\ln 1/{\varepsilon }_{*}$, with ${\varepsilon }_{*}$ belonging to the respective expected values ${\mu }_{{\widehat{D}}_0}$ (the solid line has a slope of 1/2). (b) Comparison of the harmonic (lower line) and arithmetic mean (upper line) estimates. For both panels $T=10l,l=1,...,50$. Discrete data points that indicate the means and standard deviations of dimension estimates are connected to guide the eye.

Figure 3

Fractal dimensions in the area-preserving closed baker map with a leak for several values of the additive perturbation strength $\delta$. Error bars indicate the standard deviation of the sample mean in the spirit of Eq. (25). An increasing number of experiments were done for increasing $\delta$ ($R=5\times {10}^3\delta$ and $R=10$ for $\delta =0$). Five different estimators are employed as described in the Appendix. All estimates are obtained for six values of $\delta \in [0,0.1]$ with 0.02 increments, but results using the different estimators for the same $\delta$ are plotted with a spacing for better visibility. A gray curve corresponding to the best fitting quadratic polynomial emphasizes the nonmonotonic dependence of the dimension on the perturbation strength.