Statistical properties for flows with unbounded roof function, including the Lorenz attractor

TL;DR

This paper proves a simplified central limit theorem and related statistical results for Lorenz attractors and similar systems, enabling better understanding of their stochastic behavior and homogenization properties.

Contribution

It provides a simplified proof of the central limit theorem for Lorenz attractors and extends results to more general singular hyperbolic attractors, including homogenization in fast-slow systems.

Findings

Central limit theorem established for Lorenz attractors

Functional CLT and moment estimates derived

Homogenization to stochastic differential equations demonstrated

Abstract

For geometric Lorenz attractors (including the classical Lorenz attractor) we obtain a greatly simplified proof of the central limit theorem which applies also to the more general class of codimension two singular hyperbolic attractors. We also obtain the functional central limit theorem and moment estimates, as well as iterated versions of these results. A consequence is deterministic homogenisation (convergence to a stochastic differential equation) for fast-slow dynamical systems whenever the fast dynamics is singularly hyperbolic of codimension two.