Bódai Tamás, Altmann Eduardo G, Endler Antonio
KlimaCampus, Institute of Meteorology, University of Hamburg, Grindelberg 5, 20144 Hamburg, Germany.
Phys Rev E Stat Nonlin Soft Matter Phys. 2013 Apr;87(4):042902. doi: 10.1103/PhysRevE.87.042902. Epub 2013 Apr 3.
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 ([proportionality]1/lnN). 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.
我们研究随机扰动对完全混沌开放系统的影响。扰动可以独立地应用于每个轨迹（白噪声），也可以同时应用于所有轨迹（随机映射）。我们通过推广开放混沌系统理论并引入一个与时间相关的条件映射不变测度来比较这两种情况。对于相同的扰动强度，我们表明随机映射的逃逸率总是大于噪声映射的逃逸率。在随机映射中，我们表明相关分形集的逃逸率κ和维度D通常非单调地依赖于随机扰动的强度。我们讨论了κ和D的有限尺寸估计量的准确性（偏差）和精度（方差），并表明估计精度随轨迹数N的提高极其缓慢（∝1/lnN）。我们还认为有限尺寸的D估计量通常存在偏差。一般理论结果与面积守恒面包师映射中的解析计算和数值模拟相结合。
