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A Mathematical Framework for Linear Response Theory for Nonautonomous Systems

Stefano Galatolo, Valerio Lucarini

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.

A Mathematical Framework for Linear Response Theory for Nonautonomous Systems

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.
Paper Structure (20 sections, 19 theorems, 140 equations)

This paper contains 20 sections, 19 theorems, 140 equations.

Table of Contents

  1. Introduction
  2. Sequential nonautonomous systems, sequence spaces, and global map
  3. Loss of memory
  4. Linear response
  5. A resolvent estimate on the stable (zero-mass) subspace
  6. Linear response and the series formula
  7. Linear response for sequential composition of expanding maps
  8. Uniform exponential loss of memory for sequential composition of expanding maps near $T_0$
  9. Perturbations of the sequential composition and verification of the Assumptions \ref{['ass:strong_derivative']} and \ref{['ass:mu_bound']}
  10. Post-composition perturbations.
  11. Uniform strong bounds for the equivariant family (Assumption \ref{['ass:mu_bound']})
  12. Strong differentiability along the equivariant family (Assumption \ref{['ass:strong_derivative']})
  13. Derivative of the diffeomorphism transfer operator.
  14. Verification of Assumption \ref{['ass:strong_derivative']}.
  15. Conclusion: linear response for sequential expanding maps with post-composition kicks
  16. ...and 5 more sections

Key Result

Proposition 3

Assume that $(L^{\varepsilon}_n)$ has loss of memory on $V_s$ in the sense of eq:lom_general. Then $\mathbb T_\varepsilon$ has at most one fixed point in $\ell^\infty(\mathbb Z;B_s)$. Equivalently, there exists at most one bounded equivariant family $\boldsymbol\mu$ with $\sup_n\|\mu_n\|_s<\infty$.

Theorems & Definitions (25)

  • Definition 1: Loss of memory on the strong space
  • Remark 2
  • Proposition 3: Uniqueness of the fixed point in $\ell^\infty$
  • Remark 4: Existence via pullback limits
  • Remark 8
  • Lemma 9: Neumann series for the global cocycle
  • Lemma 10: $\varepsilon log(\varepsilon)$-Continuity of the resolvent in mixed norm
  • Theorem 11: Linear response via the global-map resolvent
  • Remark 12: Connection with the "resolvent of $F$"
  • Lemma 13: Unif. exponential loss of memory for seq. expanding maps.
  • ...and 15 more