Calculation of Lyapunov Exponents using Nonstandard Finite Difference Discretization Scheme: A Case Study

Abstract

Stability analysis of nonsmooth systems, using Lyapunov direct method, is a difficult task. An alternative way of studying the stability in such systems is to employ the concept of Lyapunov exponents. However, two difficulties in the calculation of Lyapunov exponents, i.e. numerical artifacts and low computational efficiency, often prohibit the application of this otherwise powerful method. In this paper, a case study of the stability analysis of a switching contact task control system using the concept of Lyapunov exponents is presented. The goal is to carry out a numerical investigation of the method of nonstandard finite difference discretization, for constructing discrete models of differential equations that describe the Lyapunov exponents for such a nonsmooth system. It is shown that, as compared with the standard fourth-order Runge-Kutta method, the nonstandard finite difference discretization scheme provides numerically stable results with a larger critical integration step-size and less computation time. Therefore, from numerical stability and computational efficiency viewpoints, the nonstandard finite difference discretization method is meritorious and should be given consideration when stability analysis of systems using Lyapunov exponents is of concern.