On Some Aspects of the Response to Stochastic and Deterministic Forcings

Abstract

The perturbation theory of operator semigroups is used to derive response formulas for a variety of combinations of acting forcings and reference background dynamics. In the case of background stochastic dynamics, we decompose the response formulas using the Koopman operator generator eigenfunctions and the corresponding eigenvalues, thus providing a functional basis towards identifying relaxation timescales and modes in physically relevant systems. To leading order, linear response gives the correction to expectation values due to extra deterministic forcings acting on either stochastic or chaotic dynamical systems. When considering the impact of weak noise, the response is linear in the intensity of the (extra) noise for background stochastic dynamics, while the second order response given the leading order correction when the reference dynamics is chaotic. In this latter case we clarify that previously published diverging results can be brought to common ground when a suitable interpretation - Stratonovich vs. Ito - of the noise is given. Finally, the response of two-point correlations to perturbations is studied through the resolvent formalism via a perturbative approach. Our results allow, among other things, to estimate how the correlations of a chaotic dynamical system changes as a results of adding stochastic forcing.

References in corpus (11)

• Fluctuation-Dissipation: Response Theory in Statistical Physics
• Applied Koopmanism
• A review of linear response theory for general differentiable dynamical systems
• An update on nonequilibrium linear response
• Response theory: a trajectory-based approach
• Linear response, susceptibility and resonances in chaotic toy models
• Parametrization of stochastic multiscale triads
• Noise-induced stabilization of collective dynamics
• Exact Response Theory and Kuramoto dynamics
• A trajectory-driven algorithm for differentiating SRB measures on unstable manifolds
• Projections of the Transient State-Dependency of Climate Feedbacks

Cited by in corpus (12)

• Multistability and Intermediate Tipping of the Atlantic Ocean Circulation
• Theoretical tools for understanding the climate crisis from Hasselmann's program and beyond
• Limits to predictability of the asymptotic state of the Atlantic Meridional Overturning Circulation in a conceptual climate model
• Detecting and Attributing Change in Climate and Complex Systems: Foundations, Green's Functions, and Nonlinear Fingerprints
• A Punctuated Equilibrium Analysis of the Climate Evolution of Cenozoic exhibits a Hierarchy of Abrupt Transitions
• Dimension reduction of noisy interacting systems
• Response Theory Identifies Reaction Coordinates and Explains Critical Phenomena in Noisy Interacting Systems
• Analysis of a two-layer energy balance model: long time behaviour and greenhouse effect
• A General Framework for Linking Free and Forced Fluctuations via Koopmanism
• Probabilistic Measures for Biological Adaptation and Resilience
• Bridging the Gap between Koopmanism and Response Theory: Using Natural Variability to Predict Forced Response
• Kolmogorov Modes and Linear Response of Jump-Diffusion Models