On the Validity of Linear Response Theory in High-Dimensional Deterministic Dynamical Systems

Abstract

This theoretical work considers the following conundrum: linear response theory is successfully used by scientists in numerous fields, but mathematicians have shown that typical low-dimensional dynamical systems violate the theory's assumptions. Here we provide a proof of concept for the validity of linear response theory in high-dimensional deterministic systems for large-scale observables. We introduce an exemplary model in which observables of resolved degrees of freedom are weakly coupled to a large, inhomogeneous collection of unresolved chaotic degrees of freedom. By employing statistical limit laws we give conditions under which such systems obey linear response theory even if all the degrees of freedom individually violate linear response. We corroborate our result with numerical simulations.

Quadratic response of random and deterministic dynamical systems

“…In the physical literature, often borrowing the point of view of statistical mechanics, linear response formulae for several kinds of stochastic systems and for several aspects of their statistical behavior have been proposed and applied in various contexts (see [20] and [10] for general surveys), notably in climate science where several applications and estimation methods have been proposed, often in relation with the understanding of the 1 Typically this is done by modelizing the evolution of the system at a small scale as a random perturbation of the large scale dynamics or, in the presence of different time scales (fast-slow systems) one can modelize the evolution of the fast component as a random perturbation of the slow one. Sometimes random dynamical system appear as a model for an "infinite dimensional limit" of deterministic dynamical systems having many interacting components (for an example related to linear response see [38]). nature of tipping points in the climate evolution (see the introduction of [19] or [28], [27], [29], [30]).…”

Linear response for macroscopic observables in high-dimensional systems

“…To determine the smoothness of E ε Ψ , we determine its Chebyshev coefficients on a Chebyshev roots grid of 1000 points. We find that the Chebyshev coefficients decay as (k −4 ), which is indicative of E ε Ψ being between C 3 − and C 4 differentiable over a large interval: this level of differentiability, as we will see below, is connected to the smoothness of the raised-cosine distribution (9), which is also C 3 [50]. We have also employed the test statistics for higher-order linear response developed in [26] to test the null-hypothesis of E ε Ψ being well-approximated by a linear combination of T k (0.1 −1 (ε + 0.1)), k = 0, .…”

“…We therefore consider here two cases: the case when ν(a) is smooth, in particular at least once-differentiable with respect to a, and the case when ν(a) is non-smooth, for example when ν(a) is a linear combination of delta functions. Similar to [50] we choose as a smooth distribution the raised cosine distribution supported on the interval [3.7, 3.8], which is given by…”

Correcting for model changes in statistical postprocessing – an approach based on response theory

“…As such, the success of this method also depends on the quality of the past postprocessing scheme. There are situations where linear response theory is known to fail, but statistical tests which allow for the identification of its breakdown have been derived in Gottwald et al (2016) and in Wormell and Gottwald (2018). In addition, the approach presented here applies only for models for which a tangent model is available.…”