Stochastic perturbations in open chaotic systems: random versus noisy maps

TL;DR

This paper compares the effects of independent random perturbations and simultaneous noisy perturbations on open chaotic systems, revealing differences in escape rates, fractal dimensions, and estimation biases through theory and simulations.

Contribution

It introduces a generalized theory for open chaotic systems under time-dependent perturbations and compares random versus noisy maps, highlighting differences in escape rates and estimation biases.

Findings

Escape rate of random maps exceeds that of noisy maps at same perturbation strength.

Fractal dimensions depend nonmonotonically on perturbation intensity.

Finite-size estimators of escape rate and dimension are biased and converge slowly.

Abstract

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.