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On the analysis of pseudo-orbits of continuous chaotic nonlinear systems simulated using discretization schemes in a digital computer

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  • Nepomuceno, Erivelton Geraldo
  • Mendes, Eduardo M.A.M.

Abstract

This paper reports the existence of more than one pseudo-orbit when simulating continuous nonlinear systems using a digital computer in a set-up different from the ones normally seen in the literature, that is, in a set-up where the step-size is not varied, the discretization scheme is kept the same as well as the initial conditions. Taking advantage of the roundoff error, a simple but effective method to determine a lower bound error and the critical time for the pseudo-orbits is used and the connection to the maximum (positive) Lyapunov exponent is established considering the bit resolution and the computational platform used for the simulations. To illustrate the effectiveness of the method and problems of using discretization schemes for simulating continuous nonlinear systems in a digital computer, the well-known Lorenz equations, the Rossler hyperchaos system, Mackey–Glass equation and the Sprott A system are used. The method can help the user of such schemes to keep track of the reliability of numerical simulations.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:chsofr:v:95:y:2017:i:c:p:21-32
    DOI: 10.1016/j.chaos.2016.12.002
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    References listed on IDEAS

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    1. Liao, Shijun, 2013. "On the numerical simulation of propagation of micro-level inherent uncertainty for chaotic dynamic systems," Chaos, Solitons & Fractals, Elsevier, vol. 47(C), pages 1-12.
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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. 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.
    3. Zhang, Xiaofang & Li, Hongqing & Jiang, Wenan & Chen, Liqun & Bi, Qinsheng, 2022. "Exploiting multiple-frequency bursting of a shape memory oscillator," Chaos, Solitons & Fractals, Elsevier, vol. 158(C).
    4. Nepomuceno, Erivelton G. & Martins, Samir A.M. & Silva, Bruno C. & Amaral, Gleison F.V. & Perc, Matjaž, 2018. "Detecting unreliable computer simulations of recursive functions with interval extensions," Applied Mathematics and Computation, Elsevier, vol. 329(C), pages 408-419.
    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.
    6. Ren, Weikai & Jin, Zhijun, 2023. "Phase space visibility graph," Chaos, Solitons & Fractals, Elsevier, vol. 176(C).
    7. Nazarimehr, Fahimeh & Rajagopal, Karthikeyan & Khalaf, Abdul Jalil M. & Alsaedi, Ahmed & Pham, Viet-Thanh & Hayat, Tasawar, 2018. "Investigation of dynamical properties in a chaotic flow with one unstable equilibrium: Circuit design and entropy analysis," Chaos, Solitons & Fractals, Elsevier, vol. 115(C), pages 7-13.
    8. De Micco, L. & Antonelli, M. & Larrondo, H.A., 2017. "Stochastic degradation of the fixed-point version of 2D-chaotic maps," Chaos, Solitons & Fractals, Elsevier, vol. 104(C), pages 477-484.
    9. Amaral, Gleison F.V. & Nepomuceno, Erivelton G., 2018. "A smooth-piecewise model to the Cord Attractor," Chaos, Solitons & Fractals, Elsevier, vol. 109(C), pages 31-35.
    10. Guedes, Priscila F.S. & Mendes, Eduardo M.A.M. & Nepomuceno, Erivelton, 2022. "Effective computational discretization scheme for nonlinear dynamical systems," Applied Mathematics and Computation, Elsevier, vol. 428(C).

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