Fast-slow partially hyperbolic systems versus Freidlin-Wentzell random systems

TL;DR

This paper investigates the behavior of fast-slow partially hyperbolic systems, demonstrating their similarity to Freidlin-Wentzell stochastic systems over certain timescales and exploring phenomena like positive Lyapunov exponents linked to foliation properties.

Contribution

It establishes a connection between deterministic fast-slow systems and stochastic models, revealing new phenomena related to Lyapunov exponents and foliation structure.

Findings

Rescaled slow variable behavior approximates Freidlin-Wentzell systems

Identification of a sink with all positive Lyapunov exponents

Link between foliation lack of absolute continuity and system dynamics

Abstract

We consider a simple class of fast-slow partially hyperbolic dynamical systems and show that the (properly rescaled) behaviour of the slow variable is very close to a Friedlin--Wentzell type random system for times that are rather long, but much shorter than the metastability scale. Also, we show the possibility of a "sink" with all the Lyapunov exponents positive, a phenomenon that turns out to be related to the lack of absolutely continuity of the central foliation.