[Paper Review] A short survey on Lyapunov dimension for finite dimensional dynamical systems in Euclidean space

Г. А. Леонов, Н. В. Кузнецов|arXiv (Cornell University)|Oct 13, 2015

Chaos control and synchronization80 references3 citations

TL;DR

This survey provides a rigorous, accessible synthesis of Lyapunov dimension theory for finite-dimensional dynamical systems in Euclidean space, connecting foundational works by Kaplan-Yorke, Douady-Oesterlé, Constantin et al., and Leonov. It presents an analytical method for estimating Lyapunov dimension using the direct Lyapunov method and derives exact formulas for well-known systems, including the Lorenz and generalized Lorenz systems with hidden attractors, validating results via numerical computation in MATLAB.

ABSTRACT

Nowadays there are a number of surveys and theoretical works devoted to the Lyapunov exponents and Lyapunov dimension, however most of them are devoted to infinite dimensional systems or rely on special ergodic properties of the system. At the same time the provided illustrative examples are often finite dimensional systems and the rigorous proof of their ergodic properties can be a difficult task. Also the Lyapunov exponents and Lyapunov dimension have become so widespread and common that they are often used without references to the rigorous definitions or pioneering works. The survey is devoted to the finite dimensional dynamical systems in Euclidean space and its aim is to explain, in a simple but rigorous way, the connection between the key works in the area: by Kaplan and Yorke (the concept of Lyapunov dimension, 1979), Douady and Oesterle (estimation of Hausdorff dimension via the Lyapunov dimension of maps, 1980), Constantin, Eden, Foias, and Temam (estimation of Hausdorff dimension via the Lyapunov exponents and dimension of dynamical systems, 1985-90), Leonov (estimation of the Lyapunov dimension via the direct Lyapunov method, 1991), and numerical methods for the computation of Lyapunov exponents and Lyapunov dimension. In this survey a concise overview of the classical results is presented, various definitions of Lyapunov exponents and Lyapunov dimension are discussed. An effective analytical method for the estimation of Lyapunov dimension is presented, its application to the self-excited and hidden attractors of well-known dynamical systems is demonstrated, and analytical formulas of exact Lyapunov dimension are obtained.

Motivation & Objective

• To clarify and unify the theoretical foundations of Lyapunov dimension and Lyapunov exponents for finite-dimensional systems in Euclidean space.
• To bridge key contributions from Kaplan-Yorke (1979), Douady-Oesterlé (1980), Constantin et al. (1985–90), and Leonov (1991) in a coherent framework.
• To present an effective analytical method for estimating Lyapunov dimension using the direct Lyapunov method.
• To derive analytical formulas for exact Lyapunov dimension in prominent dynamical systems, including those with self-excited and hidden attractors.
• To provide a practical MATLAB implementation for computing finite-time Lyapunov exponents and Lyapunov dimension, with validation on benchmark systems.

Proposed method

• The survey establishes connections between the Hausdorff dimension and Lyapunov dimension via the singular value function and the Jacobian matrix of a map.
• It applies the Douady–Oesterlé theorem to justify the use of the Kaplan–Yorke formula for finite-time Lyapunov exponents.
• The direct Lyapunov method is used to derive analytical estimates of the Lyapunov dimension, avoiding reliance on ergodic theory.
• A numerical algorithm is implemented in MATLAB to compute finite-time Lyapunov exponents via variational equations and QR/SVD factorization of the fundamental matrix.
• The method uses a product SVD algorithm to iteratively update the fundamental matrix and compute Lyapunov exponents from logarithmic growth rates of singular values.
• The Lyapunov dimension is computed using the Kaplan–Yorke formula applied to sorted finite-time Lyapunov exponents.

Experimental results

Research questions

• RQ1 How can the Lyapunov dimension be rigorously defined and connected to the Hausdorff dimension for finite-dimensional dynamical systems?
• RQ2 What is the theoretical justification for applying the Kaplan–Yorke formula to finite-time Lyapunov exponents?
• RQ3 Can analytical formulas for the exact Lyapunov dimension be derived for well-known systems such as the Lorenz and generalized Lorenz systems?
• RQ4 How does the direct Lyapunov method enable estimation of the Lyapunov dimension without requiring ergodic properties?
• RQ5 What is the accuracy and reliability of numerical computation of Lyapunov exponents and dimension using MATLAB-based algorithms?

Key findings

• The survey establishes that the Lyapunov dimension provides an upper bound for the Hausdorff dimension of attractors in finite-dimensional systems.
• For the generalized Lorenz system with parameters r=700, σ=4, b=1, a=0.0052, the Lyapunov exponents are computed as approximately 1.873, 0.000, and -11.873, yielding a Lyapunov dimension of 2.158.
• Analytical formulas for the exact Lyapunov dimension are derived for the Henon map, Lorenz system, Glukhovsky-Dolzhansky system, Tigan and Yang systems, and Shimizu-Morioka system.
• The direct Lyapunov method enables rigorous estimation of the Lyapunov dimension without assuming ergodicity or statistical properties of the system.
• The numerical implementation in MATLAB successfully computes finite-time Lyapunov exponents and dimension, with convergence verified over long integration times.
• The method confirms that the Lyapunov dimension is invariant under diffeomorphisms, supporting its use as a robust geometric invariant.