Disentangling multi-level systems: averaging, correlations and memory

Abstract

We consider two weakly coupled systems and adopt a perturbative approach based on the Ruelle response theory to study their interaction. We propose a systematic way to parametrize the effect of the coupling as a function of only the variables of a system of interest. Our focus is on describing the impacts of the coupling on the long-term statistics rather than on the finite-time behaviour. By direct calculation, we find that, at first order, the coupling can be surrogated by adding a deterministic perturbation to the autonomous dynamics of the system of interest. At second order, there are additionally two separate and very different contributions. One is a term taking into account the second order contributions of the fluctuations in the coupling, which can be parametrized as a stochastic forcing with given spectral properties. The other one is a memory term, coupling the system of interest to its previous history, through the correlations of the second system. If these correlations are known, this effect can be implemented as a perturbation with memory on the single system. In order to treat this case, we present an extension to Ruelle's response theory able to deal with integral operators. We discuss our results in the context of other methods previously proposed to disentangle the dynamics of two coupled systems. We emphasize that our results do not rely on assuming a time scale separation, and, if such a separation exist, can be used equally well to study the statistics of the slow as well as that of the fast variables. By recursively applying the technique proposed here, we can treat the general case of multilevel systems.

References in corpus (3)

• A review of linear response theory for general differentiable dynamical systems
• Linear response, susceptibility and resonances in chaotic toy models
• Nonequilibrium Response from the dissipative Liouville Equation

Cited by in corpus (34)

• The Physics of Climate Variability and Climate Change
• Stochastic Parameterization: Towards a new view of Weather and Climate Models
• Stochastic Climate Theory and Modelling
• Extremes and Recurrence in Dynamical Systems
• Mathematical and Physical Ideas for Climate Science
• Edge States in the Climate System: Exploring Global Instabilities and Critical Transitions
• Predicting Climate Change using Response Theory: Global Averages and Spatial Patterns
• Multi-level Dynamical Systems: Connecting the Ruelle Response Theory and the Mori-Zwanzig Approach
• Statistical and Dynamical Properties of Covariant Lyapunov Vectors in a Coupled Atmosphere-Ocean Model - Multiscale Effects, Geometric Degeneracy, and Error Dynamics
• A proof of concept for scale-adaptive parameterizations: the case of the Lorenz '96 model
• Reduced-Order Models for Coupled Dynamical Systems: Data-driven Methods and the Koopman Operator
• Data-adaptive harmonic spectra and multilayer Stuart-Landau models
• Beyond the linear Fluctuation-Dissipation Theorem: the Role of Causality
• Theoretical tools for understanding the climate crisis from Hasselmann's program and beyond
• Applications of large deviation theory in geophysical fluid dynamics and climate science
• Variational approach to closure of nonlinear dynamical systems: Autonomous case
• Ruelle-Pollicott Resonances of Stochastic Systems in Reduced State Space. Part I: Theory
• Parametrization of stochastic multiscale triads
• Stochastic parameterization of subgrid-scale processes in coupled ocean-atmosphere systems: Benefits and limitations of response theory
• Elements of a unified framework for response formulae
• Conditional Gaussian Nonlinear System: a Fast Preconditioner and a Cheap Surrogate Model For Complex Nonlinear Systems
• Lyapunov analysis of multiscale dynamics: The slow bundle of the two-scale Lorenz 96 model
• Response Formulae for $n$-point Correlations in Statistical Mechanical Systems and Application to a Problem of Coarse Graining
• Comparison of stochastic parameterizations in the framework of a coupled ocean-atmosphere model
• Dimension reduction of noisy interacting systems
• Sample Paths Estimates for Stochastic Fast-Slow Systems driven by Fractional Brownian Motion
• Edgeworth expansions for slow-fast systems with finite time scale separation
• Stochastic Climate Theory
• Rough basin boundaries in high dimension: Can we classify them experimentally?
• Effects of Stochastic Parametrization on Extreme Value Statistics
• Correcting for Model Changes in Statistical Postprocessing -- An approach based on Response Theory
• Stochastic subgrid-scale parameterization for one-dimensional shallow water dynamics using stochastic mode reduction
• Introduction to the Special Issue on the Statistical Mechanics of Climate
• A Statistical Mechanical Approach for the Parametrization of the Coupling in a Fast-Slow System