- 11-32 Valerio Lucarini
- Stochastic perturbations to dynamical systems: a response theory approach
Mar 2, 11
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Abstract. We study the impact of stochastic perturbations to deterministic dynamical systems using the formalism of the Ruelle response theory. 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 twice the square of the intensity 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 real and imaginary part of the susceptibility and thus deduce the Green function of the system for any desired observable. We provide a example of application of our results by considering the spatially extended chaotic 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.