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Synchronization and averaging in partially hyperbolic systems with fast and slow variables

Federico Bonetto, Guido Gentile

TL;DR

This work analyzes a class of skew-product dynamical systems coupling a chaotic Anosov map on $\mathbb{T}^2$ with a slow neutral dynamics on $\mathbb{T}$ through a weak dissipative interaction. It proves synchronization to an attracting invariant manifold ${\mathcal W}$ and constructs a Hölder conjugation to the linearized dynamics, yielding a unique exponentially mixing physical measure. In the small-coupling regime, the slow variable obeys an averaged differential equation, with deviations from the averaged flow controlled in probability for arbitrarily long times; the authors develop a detailed averaging framework, including correlation inequalities and auxiliary models, to rigorously justify the scaling limit and continuous-time convergence. The results illuminate fast-slow behavior in partially hyperbolic systems and provide a rigorous foundation for averaging in deterministic chaotic contexts, potentially guiding extensions to more general couplings and applications in physics. The approach combines explicit invariant-manifold construction, explicit conjugations, and sophisticated probabilistic estimates to bridge deterministic chaos with averaged dynamics.

Abstract

We study a family of dynamical systems obtained by coupling an Anosov map on the two-dimensional torus -- the chaotic system -- with the identity map on the one-dimensional torus -- the neutral system -- through a dissipative interaction. We show that the two systems synchronize: the trajectories evolve toward an attracting invariant manifold, and the full dynamics is conjugated to its linearization around the invariant manifold. As a byproduct, we obtain that there exists a unique exponentially mixing physical measure. When the interaction is small, the evolution of the variable which describes the neutral system is very close to the identity; hence, it appears as a slow variable with respect to the variable which describes the chaotic system, and which is wherefore named the fast variable. We demonstrate that, seen on a suitably long time scale, the slow variable effectively follows the solution of a deterministic differential equation obtained by averaging over the fast variable. More precisely, we prove that the invariant manifold is in probability close to the fixed point of the averaged dynamics and that the difference between the exact evolution of the slow variable, seen from the invariant manifold, and its averaged evolution is in probability exponentially decreasing for arbitrarily large times.

Synchronization and averaging in partially hyperbolic systems with fast and slow variables

TL;DR

This work analyzes a class of skew-product dynamical systems coupling a chaotic Anosov map on with a slow neutral dynamics on through a weak dissipative interaction. It proves synchronization to an attracting invariant manifold and constructs a Hölder conjugation to the linearized dynamics, yielding a unique exponentially mixing physical measure. In the small-coupling regime, the slow variable obeys an averaged differential equation, with deviations from the averaged flow controlled in probability for arbitrarily long times; the authors develop a detailed averaging framework, including correlation inequalities and auxiliary models, to rigorously justify the scaling limit and continuous-time convergence. The results illuminate fast-slow behavior in partially hyperbolic systems and provide a rigorous foundation for averaging in deterministic chaotic contexts, potentially guiding extensions to more general couplings and applications in physics. The approach combines explicit invariant-manifold construction, explicit conjugations, and sophisticated probabilistic estimates to bridge deterministic chaos with averaged dynamics.

Abstract

We study a family of dynamical systems obtained by coupling an Anosov map on the two-dimensional torus -- the chaotic system -- with the identity map on the one-dimensional torus -- the neutral system -- through a dissipative interaction. We show that the two systems synchronize: the trajectories evolve toward an attracting invariant manifold, and the full dynamics is conjugated to its linearization around the invariant manifold. As a byproduct, we obtain that there exists a unique exponentially mixing physical measure. When the interaction is small, the evolution of the variable which describes the neutral system is very close to the identity; hence, it appears as a slow variable with respect to the variable which describes the chaotic system, and which is wherefore named the fast variable. We demonstrate that, seen on a suitably long time scale, the slow variable effectively follows the solution of a deterministic differential equation obtained by averaging over the fast variable. More precisely, we prove that the invariant manifold is in probability close to the fixed point of the averaged dynamics and that the difference between the exact evolution of the slow variable, seen from the invariant manifold, and its averaged evolution is in probability exponentially decreasing for arbitrarily large times.
Paper Structure (63 sections, 53 theorems, 336 equations)

This paper contains 63 sections, 53 theorems, 336 equations.

Table of Contents

  1. Introduction
  2. Deterministic description: synchronization
  3. Probabilistic description: averaging
  4. More general settings: a brief overview of possible applications
  5. Model and Results
  6. Basic Ingredients
  7. The model
  8. Synchronization
  9. The invariant manifold
  10. The linearized map and the conjugation
  11. The physical measure
  12. Averaging
  13. Heuristic discussion
  14. Synchronization in the scaling regime
  15. The averaged map
  16. ...and 48 more sections

Key Result

Proposition 2.2

Let the functions $g_+$ and $g_-$ be, respectively, in ${\mathfrak B}_{\alpha}^+(\mathds{T}^2,\mathds{R})$ and in ${\mathfrak B}_{\alpha}^-(\mathds{T}^2,\mathds{R})$, for some $\alpha\in(0,1]$. Then for all $n\in\mathds{N}$ one has for a suitable positive constant $C_{0}$ independent of $n$, $\alpha$, $g_-$ and $g_+$.

Theorems & Definitions (114)

  • Remark 2.1
  • Proposition 2.2
  • Remark 2.3
  • Remark 2.4
  • Lemma 2.5
  • Remark 2.6
  • Remark 2.7
  • Remark 2.8
  • Theorem 1: Synchronization
  • Remark 2.9
  • ...and 104 more