Eur. Phys. J. Special Topics 229, 2167–2181 (2020)
https://doi.org/10.1140/epjst/e2020-900275-x

Regular Article

Fast and simple Lyapunov Exponents estimation in discontinuous systems

Division of Dynamics, Lodz University of Technology, ul. Stefanowskiego 1/15 Lodz, Poland

^a e-mail: marek.balcerzak.1@p.lodz.pl

Received: 6 December 2019
Accepted: 8 June 2020
Published online: 28 September 2020

Abstract

Typically, to estimate the whole spectrum of n Lyapunov Exponents (LEs), it is necessary to integrate n perturbations and to orthogonalize them. Recently it has been shown that complexity of calculations can be reduced for smooth systems: integration of (n-1) perturbations is sufficient. In this paper authors demonstrate how this simplified approach can be adopted to non-smooth or discontinuous systems. Apart from the reduced complexity, the assets of the presented approach are simplicity and ease of implementation. The paper starts with a short review of properties of LEs and methods of their estimation for smooth and non-smooth systems. Then, the algorithm of reduced complexity for smooth systems is shortly introduced. Its adaptation to non-smooth systems is described in details. Application of the method is presented for an impact oscillator. Implementation of the novel algorithm is comprehensively explained. Results of simulations are presented and validated. It is expected that the presented method can simplify investigations of non-smooth dynamical systems and support research in this field.