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Algebraic Method for the Reconstruction of Partially Observed Nonlinear Systems Using Differential and Integral Embedding

Author

Listed:
  • Artur Karimov

    (Youth Research Institute, Saint Petersburg Electrotechnical University “LETI”, Saint Petersburg 197376, Russia)

  • Erivelton G. Nepomuceno

    (Control and Modelling Group (GCOM), Department of Electrical Engineering, Federal University of São, João del-Rei, São João del-Rei MG 36307-352, Brazil)

  • Aleksandra Tutueva

    (Department of Computer Aided Design, Saint Petersburg Electrotechnical University “LETI”, Saint Petersburg 197376, Russia)

  • Denis Butusov

    (Youth Research Institute, Saint Petersburg Electrotechnical University “LETI”, Saint Petersburg 197376, Russia)

Abstract

The identification of partially observed continuous nonlinear systems from noisy and incomplete data series is an actual problem in many branches of science, for example, biology, chemistry, physics, and others. Two stages are needed to reconstruct a partially observed dynamical system. First, one should reconstruct the entire phase space to restore unobserved state variables. For this purpose, the integration or differentiation of the observed data series can be performed. Then, a fast-algebraic method can be used to obtain a nonlinear system in the form of a polynomial dynamical system. In this paper, we extend the algebraic method proposed by Kera and Hasegawa to Laurent polynomials which contain negative powers of variables, unlike ordinary polynomials. We provide a theoretical basis and experimental evidence that the integration of a data series can give more accurate results than the widely used differentiation. With this technique, we reconstruct Lorenz attractor from a one-dimensional data series and B. Muthuswamy’s circuit equations from a three-dimensional data series.

Suggested Citation

  • Artur Karimov & Erivelton G. Nepomuceno & Aleksandra Tutueva & Denis Butusov, 2020. "Algebraic Method for the Reconstruction of Partially Observed Nonlinear Systems Using Differential and Integral Embedding," Mathematics, MDPI, vol. 8(2), pages 1-22, February.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:2:p:300-:d:324487
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    References listed on IDEAS

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    1. Luis A. Aguirre & Christophe Letellier, 2009. "Modeling Nonlinear Dynamics and Chaos: A Review," Mathematical Problems in Engineering, Hindawi, vol. 2009, pages 1-35, June.
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    Cited by:

    1. Ostrovskii, Valerii Yu. & Rybin, Vyacheslav G. & Karimov, Artur I. & Butusov, Denis N., 2022. "Inducing multistability in discrete chaotic systems using numerical integration with variable symmetry," Chaos, Solitons & Fractals, Elsevier, vol. 165(P1).
    2. Karimov, Artur & Kopets, Ekaterina & Karimov, Timur & Almjasheva, Oksana & Arlyapov, Viacheslav & Butusov, Denis, 2023. "Empirically developed model of the stirring-controlled Belousov–Zhabotinsky reaction," Chaos, Solitons & Fractals, Elsevier, vol. 176(C).

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