Quadratic response of random and deterministic dynamical systems

Abstract

We consider the linear and quadratic higher order terms associated to the response of the statistical properties of a dynamical system to suitable small perturbations. These terms are related to the first and second derivative of the stationary measure with respect to the change of some parameters, expressing how the statistical properties of the system varies under the perturbation. We show a general framework in which one can obtain rigorous convergence and formulas for these two terms. The framework is flexible enough to be applied both to deterministic and random systems. We give examples of such an application computing linear and quadratic response for Arnold maps with additive noise and deterministic expanding maps.

28 pages. Electronic copy of final peer-reviewed manuscript accepted for publication in Chaos

References in corpus (11)

• An update on nonequilibrium linear response
• A simple framework to justify linear response theory
• Mathematical and Physical Ideas for Climate Science
• Revising and Extending the Linear Response Theory for Statistical Mechanical Systems: Evaluating Observables as Predictors and Predictands
• Stochastic perturbations to dynamical systems: a response theory approach
• On the validity of linear response theory in high-dimensional deterministic dynamical systems
• Controlling the statistical properties of expanding maps
• A regularity result for fixed points, with applications to linear response
• Optimal linear responses for Markov chains and stochastically perturbed dynamical systems
• Response Formulae for $n$-point Correlations in Statistical Mechanical Systems and Application to a Problem of Coarse Graining
• Statistical properties of dynamics. Introduction to the functional analytic approach

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