The Lyapunov dimension and its computation for self-excited and hidden attractors in the Glukhovsky-Dolzhansky fluid convection model

TL;DR

This paper rigorously defines and computes the Lyapunov dimension for attractors in the Glukhovsky-Dolzhansky fluid model, discussing differences between self-excited and hidden attractors, with a tutorial on numerical estimation.

Contribution

It provides a rigorous approach for calculating the Lyapunov dimension and offers an exact formula for the global attractors in the model, including a tutorial example.

Findings

Exact Lyapunov dimension formula derived

Differences in estimation for self-excited and hidden attractors discussed

Numerical estimation tutorial provided

Abstract

Consideration of various hydrodynamic phenomena involves the study of the Navier-Stokes (N-S) equations, what is hard enough for analytical and numerical investigations since already in three-dimensional (3D) case it is a challenging task to study the limit behavior of N-S solutions. The low-order models (LOMs) derived from the initial N-S equations by Galerkin method allow one to overcome difficulties in studying the limit behavior and existence of attractors. Among the simple LOMs with chaotic attractors there are famous Lorenz system, which is an approximate model of two-dimensional convective flow and Glukhovsky-Dolzhansky model, which describes a convective process in three-dimensional rotating fluid and can be considered as an approximate model of the World Ocean. One of the widely used dimensional characteristics of attractors is the Lyapunov dimension. In the study we follow a rigorous approach for the definition of the Lyapunov dimension and justification of its computation by the Kaplan-Yorke formula, without using statistical physics assumptions. The exact Lyapunov dimension formula for the global attractors is obtained and peculiarities of the Lyapunov dimension estimation for self-excited and hidden attractors are discussed. A tutorial on numerical estimation of the Lyapunov dimension on the example of the Glukhovsky-Dolzhansky model is presented.