2011/03/31 by Valerio Lucarini · 1 citation
Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Chaos control and synchronization #Dynamical systems theory #Invariant measure #Linear system #Multiplicative function #Multiplicative noise #Observable #Perturbation (astronomy) #Spectral density #White noise #cond-mat.stat-mech #nlin.AO #nlin.CD #physics.ao-ph #physics.comp-ph #physics.flu-dyn #stochastic dynamics and bifurcation
paper · pdf · doi:10.1007/s10955-012-0422-0
18 pages, 2 figures [updated version]
arxiv created 2011/09/12 · openalex publication_date 2012/01/18 · arxiv updated 2015/05/27 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
We study the impact of stochastic perturbations to deterministic dynamical systems using the formalism of the Ruelle response theory and explore how stochastic noise can be used to explore the properties of the underlying deterministic dynamics of a system. We find the expression for the change in the expectation value of a general observable when a white noise forcing is introduced in the system, both in the case of additive and multiplicative noise. We also show that the difference between the expectation value of the power spectrum of an observable in the stochastically perturbed case and of the same observable in the unperturbed case is equal to the variance of the noise times the square of the modulus of the susceptibility describing the frequency-dependent response of the system to perturbations with the same spatial patterns as the considered stochastic forcing. Using Kramers-Kronig theory, it is then possible to derive the susceptibility and thus deduce the Green function of the system for any desired observable. We then extend our results to rather general patterns of random forcing, from the case of several white noise forcings, to noise terms with memory, up to the case of a space-time random field. Explicit formulas are provided for each relevant case. As a general result, we find, using an argument of positive-definiteness, that the power spectrum of the stochastically perturbed system is larger at all frequencies than the power spectrum of the unperturbed system. We provide a example of application of our results by considering the Lorenz 96 model. These results clarify the property of stochastic stability of SRB measures in Axiom A flows, provide tools for analysing stochastic parameterisations and related closure ansatz to be implemented in modelling studies, and introduce new ways to study the response of a system to external perturbations.