The Lyapunov dimension formula for the global attractor of the Lorenz system

TL;DR

This paper extends the analytical formula for the Lyapunov dimension of the Lorenz system's attractor to a broader parameter range, including physically relevant cases with mixed stability equilibria.

Contribution

It proves the Lyapunov dimension formula's validity for a wider set of parameters, enhancing understanding of chaotic attractors in the Lorenz system.

Findings

Lyapunov dimension formula now valid for all hyperbolically unstable equilibria parameters.

Extended the class of Lyapunov-type functions used in the analysis.

Applicable to cases with one unstable and two stable equilibria.

Abstract

The exact Lyapunov dimension formula for the Lorenz system has been analytically obtained first due to G.A.Leonov in 2002 under certain restrictions on parameters, permitting classical values. He used the construction technique of special Lyapunov-type functions developed by him in 1991 year. Later it was shown that the consideration of larger class of Lyapunov-type functions permits proving the validity of this formula for all parameters of the system such that all the equilibria of the system are hyperbolically unstable. In the present work it is proved the validity of the formula for Lyapunov dimension for a wider variety of parameters values, which include all parameters satisfying the classical physical limitations. One of the motivation of this work is the possibility of computing a chaotic attractor in the Lorenz system in the case of one unstable and two stable equilibria.