Construction and description of the stationary measure of weakly dissipative dynamical systems

TL;DR

This paper analyzes how small dissipation affects the stationary measure in dynamical systems, revealing singularities and deriving statistical properties, correlation functions, and fractal dimensions of the attractor.

Contribution

It provides a method to handle the singular perturbation caused by small dissipation and describes the resulting measure's statistics and dimensions.

Findings

The measure becomes singular with small dissipation.

Correlation functions of the measure are derived.

The fractal and information dimensions are characterized.

Abstract

We consider the stationary measure of the dissipative dynamical system in a finite volume. A finite dissipation, however small, generally makes the measure singular, while at zero dissipation the measure is constant. Thus dissipative part of the dynamics is a singular perturbation producing an infinite change in the measure. This is a result of the infinite time of evolution that enhances the small effects of dissipation to form singularities. We show how to deal with the singularity of the perturbation and describe the statistics of the measure. We derive all the correlation functions and the statistics of "mass" contained in a small ball. The spectrum of dimensions of the attractor is obtained. The fractal dimension is equal to the space dimension, while the information dimension is equal to the Kaplan-Yorke dimension.