The Lyapunov dimension and its estimation via the Leonov method

TL;DR

This paper reviews the Lyapunov dimension, focusing on the analytical Leonov method for estimating it without phase space localization, discussing its invariance, connection to key theories, and an extension to discrete systems.

Contribution

It provides a comprehensive survey of the Leonov method, its invariance properties, and introduces an analog for discrete systems, linking it to foundational works in the field.

Findings

Leonov method allows estimation of Lyapunov dimension without localization.

Invariance of Lyapunov dimension under diffeomorphisms is discussed.

An analog of the Leonov method for discrete systems is proposed.

Abstract

Along with widely used numerical methods for estimating and computing the Lyapunov dimension there is an effective analytical approach, proposed by G.A. Leonov in 1991. The Leonov method is based on the direct Lyapunov method with special Lyapunov-like functions. The advantage of this method is that it allows one to estimate the Lyapunov dimension of invariant set without local- ization of the set in the phase space and in many cases get effectively exact Lyapunov dimension formula. In this survey the invariance of Lyapunov dimension with respect to diffeomorphisms and its connection with the Leonov method are discussed. An analog of Leonov method for discrete time dynamical systems is suggested. In a simple but rigorous way, here it is presented the connection between the Leonov method and the key related works in the area: by Kaplan and Yorke (the concept of Lyapunov dimension, 1979), Douady and Oesterle (upper bounds of Hausdorff dimension via the Lyapunov dimension of maps, 1980), Constantin, Eden, Foias, and Temam (upper bounds of Hausdorff dimension via the Lyapunov exponents and dimension of dynamical systems, 1985-90), and the numerical calculation of the Lyapunov exponents and dimension.