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Liouvillian Integrability and the Poincaré Problem for Nonlinear Oscillators with Quadratic Damping and Polynomial Forces

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Abstract

The upper bound on the degrees of irreducible Darboux polynomials associated to the ordinary differential equations \( x_{tt}+\varepsilon {x_{t}}^{2}+\eta x_{t}+f(x)=0 \) with \( f(x)\in \mathbb {C}[x]\setminus \mathbb {C} \) and ε ≠ 0 is derived. The availability of this bound provides the solution of the Poincaré problem. Results on uniqueness and existence of Darboux polynomials are presented. The problem of Liouvillian integrability for related dynamical systems is solved completely. It is proved that Liouvillian first integrals exist if and only if η = 0.

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Acknowledgments

We would like to thank the reviewers for their helpful comments and suggestions that greatly contributed to improving the final version of the article.

Funding

This research was supported by Russian Science Foundation grant 19–71–10003.

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Authors and Affiliations

  1. Department of Applied Mathematics, National Research University Higher School of Economics, 34 Tallinskaya Street, Moscow, 123458, Russian Federation

    Maria V. Demina & Nikolai S. Kuznetsov

  2. Faculty of Computer Science, National Research University Higher School of Economics, 11 Pokrovsky Boulevard, Moscow, 109028, Russian Federation

    Nikolai S. Kuznetsov

Authors
  1. Maria V. Demina
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  2. Nikolai S. Kuznetsov
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Correspondence to Maria V. Demina.

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Demina, M.V., Kuznetsov, N.S. Liouvillian Integrability and the Poincaré Problem for Nonlinear Oscillators with Quadratic Damping and Polynomial Forces. J Dyn Control Syst 27, 403–415 (2021). https://doi.org/10.1007/s10883-020-09513-2

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  • DOI: https://doi.org/10.1007/s10883-020-09513-2

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