Stochastic perturbations in open chaotic systems: Random versus noisy maps

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 κ 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 κ and D, and show that the improvement of the precision of the estimations with the number of trajectories N is extremely slow (∝ 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.

Memory effects in chaotic advection of inertial particles

“…The idea proved to be particularly well suited for understanding the advection of passive particles in random flows [71][72][73], and explains experimental findings [74]. More recently, snapshot attractors have turned out to be promising tools to describe the variability of climate dynamics in a novel way [70,[75][76][77][78]. In these settings the non-autonomous driving is typically a random noise or a sustained chaotic process.…”

Local and global approaches to the problem of Poincaré recurrences. Applications in nonlinear dynamics

“…In the framework of this problem, the role of the size and position of the leak as well as of specific features of the dynamics has been analyzed [42] and the effect of noise on the escape rate has been studied [43,44,45]. In particular, the non-monotonic dependency of the mean escape rate on the noise intensity has been reported in [45,46].…”

Diffusion for ensembles of standard maps

“…This motivates calculation of various scenarios and classification of "noisy" phenomena. A characterization of this type was introduced by Romeiras, Grebogi and Ott [15] (see also [16] In the present work we apply noise like in problem 1, of small variance, and in some cases we consider a random map like in problem 2. The processes we study can be classified into two types:…”