Stochastic perturbations in open chaotic systems: random versus noisy maps

摘要

We investigate the effects of random perturbations on fully chaotic opensystems. Perturbations can be applied to each trajectory independently (whitenoise) or simultaneously to all trajectories (random map). We compare these twoscenarios by generalizing the theory of open chaotic systems and introducing atime-dependent conditionally-map-invariant measure. For the same perturbationstrength we show that the escape rate of the random map is always larger thanthat of the noisy map. In random maps we show that the escape rate $\kappa$ anddimensions $D$ of the relevant fractal sets often depend nonmonotonically onthe intensity of the random perturbation. We discuss the accuracy (bias) andprecision (variance) of finite-size estimators of $\kappa$ and $D$, and showthat the improvement of the precision of the estimations with the number oftrajectories $N$ is extremely slow ($\propto 1/\ln N$). We also argue that thefinite-size $D$ estimators are typically biased. General theoretical resultsare combined with analytical calculations and numerical simulations inarea-preserving baker maps.