A Mathematical Framework for Linear Response Theory for Nonautonomous Systems

TL;DR

This paper develops a rigorous linear response theory for a broad class of nonautonomous systems, enabling prediction of how small external forcings influence their time-dependent statistical properties.

Contribution

It extends existing autonomous response theories to nonautonomous systems with assumptions like rapid memory loss, using a fixed-point approach on an extended measure space.

Findings

Provides explicit formulas for small perturbation effects on nonautonomous systems.

Applies the theory to compositions of expanding maps and noisy random maps.

Demonstrates exponential loss of memory under certain conditions.

Abstract

Linear Response theory aims to predict how added forcing alters the statistical properties of an unforced system. These kinds of questions have been studied predominantly for autonomous dynamical systems, yet many systems in the physical, natural, and social sciences are inherently nonautonomous, evolving in time under external forcings of various kinds (a canonical example being the climate system). In such settings, one would like to understand how the system's time dependent statistical properties change when additional infinitesimal forcings are applied. This question is of clear practical relevance, but from a rigorous mathematical viewpoint it has been addressed only for a few specific classes of systems/perturbations. Here we provide a rigorous linear response theory for a rather general class of deterministic and random nonautonomous systems satisfying a specific set of assumptions that in some sense extend the standard assumptions used in the autonomous setting. A central ingredient is rapid loss of memory, i.e. sufficiently fast forgetting of initial conditions along the nonautonomous evolution. Our main strategy is to reformulate the sequential dynamics as a fixed-point problem for a global transfer operator acting on an extended sequence space of measures. This yields explicit and readily implementable response formulas for predicting the effect of small perturbations on time-dependent statistical states. We illustrate the theory on two representative classes: sequential compositions of C3 expanding maps and sequential compositions of noisy random maps, where uniform positivity of the noise induces exponential loss of memory.