Spectrum of Lyapunov exponents in non-smooth systems evaluated using orthogonal perturbation vectors

Division of Dynamics, Lodz University of Technology, Stefanowskiego 1/15, 90-924 Lodz, Poland

^* Corresponding author: Andrzej.Stefanski@p.lodz.pl

Abstract

This paper covers application of the novel method of Lyapunov exponents (LEs) spectrum estimation in non smooth mechanical systems. In the presented method, LEs are obtained from a Poincaré map. By analysing the map instead of the full trajectory, problems with transition of perturbations through discontinuities can be avoided. However, the explicit formula of the map is usually not known. Therefore, the Jacobi matrix of the map is estimated using small perturbations of the initial point. In such a manner, direct calculation of the Jacobi matrix can be avoided. The article provides a detailed description of the method accompanied by clear schemes. The algorithm of Jacobi matrix estimation is elaborated and an example is given. Efficiency of the method is confirmed by a numerical experiment. The mechanical oscillator with impact has been simulated. Bifurcation diagrams and Lyapunov exponents graphs have been generated. It has been shown that the method provides values of the whole Lyapunov exponents spectrum with high accuracy.