[Paper Review] Averaging and rates of averaging for uniform families of deterministic fast-slow skew product systems

Alexey Korepanov, Zemer Kosloff+1 more|arXiv (Cornell University)|Sep 22, 2015

Mathematical Dynamics and Fractals27 references3 citations

TL;DR

This paper establishes averaging and rates of averaging for uniform families of deterministic fast-slow skew product systems in both discrete and continuous time, under weak statistical stability assumptions on the fast dynamics. It proves convergence to an averaged system with optimal rate $ O(\epsilon^{1/2-}) $, extending classical averaging theory beyond uniformly hyperbolic settings to include intermittent, unimodal, and Viana maps.

ABSTRACT

We consider families of fast-slow skew product maps of the form \begin{align*} x_{n+1} = x_n+εa(x_n,y_n,ε), \quad y_{n+1} = T_εy_n, \end{align*} where $T_ε$ is a family of nonuniformly expanding maps, and prove averaging and rates of averaging for the slow variables $x$ as $ε o0$. Similar results are obtained also for continuous time systems \begin{align*} \dot x = εa(x,y,ε), \quad \dot y = g_ε(y). \end{align*} Our results include cases where the family of fast dynamical systems consists of intermittent maps, unimodal maps (along the Collet-Eckmann parameters) and Viana maps.

Motivation & Objective

• To extend classical averaging theory to fast-slow skew product systems where the fast dynamics is nonuniformly expanding, rather than uniformly hyperbolic.
• To establish convergence of the slow variables to an averaged system under weak regularity assumptions on the invariant measures, replacing linear response with statistical stability.
• To derive quantitative rates of averaging, particularly the essentially optimal rate $ O(\epsilon^{1/2-}) $, in both discrete and continuous time settings.
• To handle cases where classical linear response fails, such as in Collet-Eckmann unimodal maps and intermittent maps with poor mixing.
• To provide a general framework applicable to a wide class of nonuniformly hyperbolic systems, including suspensions over such maps.

Proposed method

• Formulate the fast-slow skew product system in discrete time as $ x_{n+1} = x_n + \epsilon a(x_n, y_n, \epsilon) $, $ y_{n+1} = T_\epsilon y_n $, with $ T_\epsilon $ nonuniformly expanding.
• Use statistical stability of the invariant measures $ \nu_\epsilon $, i.e., weak convergence $ \nu_\epsilon \to \nu_0 $, in place of linear response for the slow dynamics.
• Introduce a comparison process $ w_n $ that approximates the true trajectory $ x_n $, with error bounded by $ \delta_\epsilon $, the measure of initial deviation.
• Apply a second-order averaging argument via a discrete-time version of the classical averaging theorem, controlling the difference between $ x_n $ and the averaged path.
• Use Gronwall-type estimates to bound the deviation between the true solution and the averaged ODE, leveraging Lipschitz continuity of the vector field $ \bar{a} $.
• Reduce the continuous-time case to the discrete-time setting by considering time-1 maps of the flow, and apply analogous estimates to derive convergence in continuous time.

Experimental results

Research questions

• RQ1 Can averaging be established for fast-slow skew systems when the fast dynamics is nonuniformly expanding, rather than uniformly hyperbolic?
• RQ2 What is the optimal rate of convergence to the averaged system in such nonuniformly hyperbolic settings?
• RQ3 How can statistical stability of invariant measures replace linear response in proving averaging theorems?
• RQ4 To what extent do the results extend to intermittent maps, unimodal maps (e.g., Collet-Eckmann), and Viana maps, where linear response fails?
• RQ5 Can the framework be adapted to continuous-time systems with nonuniformly hyperbolic fast flows?

Key findings

• The paper proves averaging for fast-slow skew systems with nonuniformly expanding fast dynamics, under the assumption of statistical stability of the invariant measures $ \nu_\epsilon $.
• The convergence rate of the slow variables to the averaged system is $ O(\epsilon^{1/2-}) $, which is essentially optimal and matches the known rate for uniformly expanding systems.
• The results hold for a broad class of systems, including intermittent maps with arbitrarily weak mixing, unimodal maps along Collet-Eckmann parameters, and Viana maps.
• The framework avoids the need for linear response by relying on statistical stability, which is more tractable and verifiable in nonuniformly hyperbolic settings.
• The continuous-time case is reduced to the discrete-time case via time-1 maps, and analogous averaging and rate results are obtained.
• A counterexample is provided showing that almost sure convergence fails in general for such families, highlighting the necessity of convergence in probability or in $ L^1 $.