Computation of a few Lyapunov exponents for continuous and discrete dynamical systems
type: Item
instance of: scholarly article
title: Computation of a few Lyapunov exponents for continuous and discrete dynamical systems
zbMATH Open document ID: 0834.65069
DOI: 10.1016/0168-9274(95)00033-Q
author: Luca Dieci
author: E. Van Vleck
published in: Applied Numerical Mathematics
publication date: 1996-03-31T00:00:00Z
review text: This paper deals with a QR-based method for computing the first few Lyapunov exponents of continuous and discrete finite-dimensional dynamical systems. The first few Lyapunov exponents are of importance in a number of applications, e.g. to detect chaotic or turbulent behaviour of systems. It appears that for a continuous dynamical system, computation of a few Lyapunov exponents by the QR method results in a nonlinear matrix equation which has a so called weakly skew-symmetric structure. Solutions of this type of equations are matrices with orthonormal columns, but the right-hand side of the system cannot be written in the form of a product of a skew-symmetric matrix and an orthogonal matrix. It implies that the known unitary integrators will not generally preserve orthogonality of the computed solution. The authors present methods for presenting orthogonality in weakly skew- symmetric systems. Moreover, they extend some of their previous results regarding the error analysis to the case of a few Lyapunov exponents. The paper also includes details of how the error in the computation of the exponents is estimated and provides numerical results for three nonlinear examples (double rotor map, the Lorenz equation, and a ring of oscillators).
Mathematics Subject Classification ID: 65L07
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 65L70
Mathematics Subject Classification ID: 34D08
Mathematics Subject Classification ID: 37D45
zbMATH DE Number: 803526
zbMATH Keywords: orthogonalization techniques
zbMATH Keywords: double rotor map
zbMATH Keywords: Lyapunov exponents
zbMATH Keywords: continuous and discrete finite-dimensional dynamical systems
zbMATH Keywords: QR method
zbMATH Keywords: nonlinear matrix equation
zbMATH Keywords: weakly skew-symmetric systems
zbMATH Keywords: numerical results
zbMATH Keywords: Lorenz equation
zbMATH Keywords: ring of oscillators
reviewed by: Andrzej Marciniak
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 803526