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Computation of the largest positive Lyapunov exponent using rounding mode and recursive least square algorithm

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  • Peixoto, Márcia L.C.
  • Nepomuceno, Erivelton G.
  • Martins, Samir A.M.
  • Lacerda, Márcio J.

Abstract

It has been shown that natural interval extensions (NIE) can be used to calculate the largest positive Lyapunov exponent (LLE). However, the elaboration of NIE are not always possible for some dynamical systems, such as those modelled by simple equations or by Simulink-type blocks. In this paper, we use rounding mode of floating-point numbers to compute the LLE. We have exhibited how to produce two pseudo-orbits by means of different rounding modes; these pseudo-orbits are used to calculate the Lower Bound Error (LBE). The LLE is the slope of the line gotten from the logarithm of the LBE, which is estimated by means of a recursive least square algorithm (RLS). The main contribution of this paper is to develop a procedure to compute the LLE based on the LBE without using the NIE. Additionally, with the aid of RLS the number of required points has been decreased. Eight numerical examples are given to show the effectiveness of the proposed technique.

Suggested Citation

  • Peixoto, Márcia L.C. & Nepomuceno, Erivelton G. & Martins, Samir A.M. & Lacerda, Márcio J., 2018. "Computation of the largest positive Lyapunov exponent using rounding mode and recursive least square algorithm," Chaos, Solitons & Fractals, Elsevier, vol. 112(C), pages 36-43.
    • Handle: RePEc:eee:chsofr:v:112:y:2018:i:c:p:36-43
    DOI: 10.1016/j.chaos.2018.04.032
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    References listed on IDEAS

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    1. Ricardo Fukasawa & Laurent Poirrier, 2017. "Numerically Safe Lower Bounds for the Capacitated Vehicle Routing Problem," INFORMS Journal on Computing, INFORMS, vol. 29(3), pages 544-557, August.
    2. Nepomuceno, Erivelton Geraldo & Mendes, Eduardo M.A.M., 2017. "On the analysis of pseudo-orbits of continuous chaotic nonlinear systems simulated using discretization schemes in a digital computer," Chaos, Solitons & Fractals, Elsevier, vol. 95(C), pages 21-32.
    3. Alicia Serfaty De Markus, 2001. "Detection of the onset of numerical chaotic instabilities by lyapunov exponents," Discrete Dynamics in Nature and Society, Hindawi, vol. 6, pages 1-8, January.
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    Cited by:

    1. Tutueva, Aleksandra V. & Nepomuceno, Erivelton G. & Karimov, Artur I. & Andreev, Valery S. & Butusov, Denis N., 2020. "Adaptive chaotic maps and their application to pseudo-random numbers generation," Chaos, Solitons & Fractals, Elsevier, vol. 133(C).
    2. Trobia, José & de Souza, Silvio L.T. & dos Santos, Margarete A. & Szezech, José D. & Batista, Antonio M. & Borges, Rafael R. & Pereira, Leandro da S. & Protachevicz, Paulo R. & Caldas, Iberê L. & Iaro, 2022. "On the dynamical behaviour of a glucose-insulin model," Chaos, Solitons & Fractals, Elsevier, vol. 155(C).
    3. Zhou, Shuang & Wang, Xingyuan, 2020. "Simple estimation method for the second-largest Lyapunov exponent of chaotic differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 139(C).
    4. Nepomuceno, Erivelton G. & Lima, Arthur M. & Arias-García, Janier & Perc, Matjaž & Repnik, Robert, 2019. "Minimal digital chaotic system," Chaos, Solitons & Fractals, Elsevier, vol. 120(C), pages 62-66.
    5. Nardo, Lucas G. & Nepomuceno, Erivelton G. & Arias-Garcia, Janier & Butusov, Denis N., 2019. "Image encryption using finite-precision error," Chaos, Solitons & Fractals, Elsevier, vol. 123(C), pages 69-78.

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