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Mathematical Problems in Engineering
Volume 2018, Article ID 6909151, 8 pages
https://doi.org/10.1155/2018/6909151
Research Article

On the Use of Interval Extensions to Estimate the Largest Lyapunov Exponent from Chaotic Data

1Control and Modelling Group (GCOM), Department of Electrical Engineering, Federal University of São João del-Rei, 36307-352 São João del-Rei, MG, Brazil
2Department of Electronic Engineering, School of Engineering, Federal University of Minas Gerais, 31270-901 Belo Horizonte, MG, Brazil

Correspondence should be addressed to Márcio J. Lacerda; rb.ude.jsfu@adrecal

Received 28 November 2017; Revised 12 January 2018; Accepted 28 January 2018; Published 4 March 2018

Academic Editor: Gisele Mophou

Copyright © 2018 Erivelton G. Nepomuceno et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A method to estimate the (positive) largest Lyapunov exponent (LLE) from data using interval extensions is proposed. The method differs from the ones available in the literature in its simplicity since it is only based on three rather simple steps. Firstly, a polynomial NARMAX is used to identify a model from the data under investigation. Secondly, interval extensions, which can be easily extracted from the identified model, are used to calculate the lower bound error. Finally, a simple linear fit to the logarithm of lower bound error is obtained and then the LLE is retrieved from it as the third step. To illustrate the proposed method, the LLE is calculated for the following well-known benchmarks: sine map, Rössler Equations, and Mackey-Glass Equations from identified models given in the literature and also from two identified NARMAX models: a chaotic jerk circuit and the tent map. In the latter, a Gaussian noise has been added to show the robustness of the proposed method.