A computable realization of Ruelle's formula for linear response of statistics in chaotic systems

Efficient computation of linear response of chaotic attractors with one-dimensional unstable manifolds

This paper presents the space-split sensitivity or the S3 algorithm to transform Ruelle's linear response formula into a well-conditioned ergodic-averaging computation. We prove a decomposition of Ruelle's formula that is differentiable on the unstable manifold, which we assume to be one-dimensional. This decomposition of Ruelle's formula ensures that one of the resulting terms, the stable contribution, can be computed using a regularized tangent equation, similar to in a non-chaotic system. The remaining term, known as the unstable contribution, is regularized and converted into an efficientl…

Ruelle spectrum of linear pseudo-Anosov maps

The Ruelle resonances of a dynamical system are spectral data describing the precise asymptotics of correlations. We classify them completely for a class of chaotic two-dimensional maps, the linear pseudo-Anosov maps, in terms of the action of the map on cohomology. As applications, we obtain a full description of the distributions which are invariant under the linear flow in the stable direction of such a linear pseudo-Anosov map, and we solve the cohomological equation for this flow.

Space-split algorithm for sensitivity analysis of discrete chaotic systems with unstable manifolds of arbitrary dimension

Accurate approximations of the change of system's output and its statistics with respect to the input are highly desired in computational dynamics. Ruelle's linear response theory provides breakthrough mathematical machinery for computing the sensitivity of chaotic dynamical systems, which enables a better understanding of chaotic phenomena. In this paper, we propose an algorithm for sensitivity analysis of discrete chaos with an arbitrary number of positive Lyapunov exponents. We combine the concept of perturbation space-splitting regularizing Ruelle's original expression together with measur…

Anosov系统中用于计算线性响应的空间分裂敏感性算法的严格论证

Rigorous justification for the space-split sensitivity algorithm to compute linear response in Anosov systems

本文为空间分裂敏感性（S3）算法提供了严格的数学论证，证明了S3分解的存在性及Ruelle线性响应公式两个分量计算的收敛性，从而为该高效计算方法在Anosov系统中的应用奠定了理论基础。

Linear response, susceptibility and resonances in chaotic toy models

We consider simple examples illustrating some new features of the linear response theory developed by Ruelle for dissipative and chaotic systems [{\em J. of Stat. Phys.} {\bf 95} (1999) 393]. In this theory the concepts of linear response, susceptibility and resonance, which are familiar to physicists, have been revisited due to the dynamical contraction of the whole phase space onto attractors. In particular the standard framework of the "fluctuation-dissipation" theorem breaks down and new resonances can show up oustside the powerspectrum. In previous papers we proposed and used new numerica…

Orbit complexity, initial data sensitivity and weakly chaotic dynamical systems

We give a definition of generalized indicators of sensitivity to initial conditions and orbit complexity (a measure of the information that is necessary to describe the orbit of a given point). The well known Ruelle-Pesin and Brin-Katok theorems, combined with Brudno's theorem give a relation between initial data sensitivity and orbit complexity that is generalized in the present work. The generalized relation implies that the set of points where the sensitivity to initial conditions is more than exponential in all directions is a 0 dimensional set. The generalized relation is then applied to …