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Fast and simple Lyapunov Exponents estimation in discontinuous systems

Authors
  • Balcerzak, M.1
  • Sagan, T.1
  • Dabrowski, A.1
  • Stefanski, A.1
  • 1 Division of Dynamics, Lodz University of Technology, ul. Stefanowskiego 1/15, Lodz, Poland , Lodz (Poland)
Type
Published Article
Journal
The European Physical Journal Special Topics
Publisher
Springer Berlin Heidelberg
Publication Date
Sep 28, 2020
Volume
229
Issue
12-13
Pages
2167–2181
Identifiers
DOI: 10.1140/epjst/e2020-900275-x
Source
Springer Nature
License
Green

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.

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