Computation of a few Lyapunov exponents for continuous and discrete dynamical systems
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).
continuous and discrete finite-dimensional dynamical systems
scientific article; zbMATH DE number 803526
scientific article