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Lyapunov Function-Based Approach to Estimate Attractors for a Dynamical System with the Polynomial Right Side

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Advanced Manufacturing Processes IV (InterPartner 2022)

Part of the book series: Lecture Notes in Mechanical Engineering ((LNME))

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Abstract

Stability analysis is an essential part of the study of the behavior of a dynamic system. Typically, this analysis includes finding the stationary points or limit cycles, determining their stability or instability, and identifying the regions of attraction (RoAs) of attractors. There are several classical methods for obtaining RoAs estimates, which may be divided into Lyapunov and non-Lyapunov methods; at the same time, due to the limitations of existing methods, the identification of a complex RoAs boundary is practically impossible, and it also leads to a high computational cost. The existing methods are quite effective for systems of the second and third orders. However, an increase in the dimension of the system or uncertain mechanical parameters leads to an exponential increase in the required calculations. In this regard, it is essential to design relatively simple algorithms in terms of the number of necessary operations and, at the same time, give acceptable from a practical point of view estimates of RoAs. The present paper deals with the problem of obtaining estimates of the domains of attraction and stability for a nonlinear dynamical system with a polynomial right-hand side. It is based on a particular procedure of polynomial Lyapunov function construction. As an example, this procedure is applied to estimate the domain of attraction for the mechanical system of two coupled oscillators.

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Author information

Authors and Affiliations

  1. Universitat Politècnica de Catalunya, 31, Carrer de Jordi Girona, 08034, Barcelona, Spain

    Volodymyr Puzyrov

  2. Nizhyn Gogol State University, 2, Grafska Street, Nizhyn, 16600, Ukraine

    Volodymyr Puzyrov & Nataliya Losyeva

  3. Universitat de Barcelona, 585, Gran Via de les Corts Catalanes, 08007, Barcelona, Spain

    Nataliya Losyeva

  4. National Aerospace University “KhAI”, 17, Chkalova Street, Kharkiv, 61070, Ukraine

    Nina Savchenko

  5. National University of Food Technologies, 68, Volodymyrska Street, Kyiv, 01033, Ukraine

    Oksana Nikolaieva

  6. Sumy State Pedagogical University, 87, Romenska Street, Sumy, 40002, Ukraine

    Olga Chashechnikova

Authors
  1. Volodymyr Puzyrov
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  2. Nataliya Losyeva
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  3. Nina Savchenko
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  4. Oksana Nikolaieva
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  5. Olga Chashechnikova
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Corresponding author

Correspondence to Nina Savchenko .

Editor information

Editors and Affiliations

  1. Odessa Polytechnic National University, Odessa, Ukraine

    Volodymyr Tonkonogyi

  2. Sumy State University, Sumy, Ukraine

    Vitalii Ivanov

  3. Poznan University of Technology, Poznan, Poland

    Justyna Trojanowska

  4. Odessa Polytechnic National University, Odessa, Ukraine

    Gennadii Oborskyi

  5. Sumy State University, Sumy, Ukraine

    Ivan Pavlenko

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Puzyrov, V., Losyeva, N., Savchenko, N., Nikolaieva, O., Chashechnikova, O. (2023). Lyapunov Function-Based Approach to Estimate Attractors for a Dynamical System with the Polynomial Right Side. In: Tonkonogyi, V., Ivanov, V., Trojanowska, J., Oborskyi, G., Pavlenko, I. (eds) Advanced Manufacturing Processes IV. InterPartner 2022. Lecture Notes in Mechanical Engineering. Springer, Cham. https://doi.org/10.1007/978-3-031-16651-8_46

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  • DOI: https://doi.org/10.1007/978-3-031-16651-8_46

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-16650-1

  • Online ISBN: 978-3-031-16651-8

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