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Regul. Chaotic Dyn., 2018, Volume 23, Issue 4, Pages 471–479 (Mi rcd334)  

This article is cited in 19 scientific papers (total in 19 papers)

Exact Solutions and Integrability of the Duffing–Van der Pol Equation

Nikolay A. Kudryashov

Department of Applied Mathematics, National Research Nuclear University MEPhI, Kashirskoe sh. 31, Moscow, 115409 Russia

Abstract: The force-free Duffing–Van der Pol oscillator is considered. The truncated expansions for finding the solutions are used to look for exact solutions of this nonlinear ordinary differential equation. Conditions on parameter values of the equation are found to have the linearization of the Duffing–Van der Pol equation. The Painlevé test for this equation is used to study the integrability of the model. Exact solutions of this differential equation are found. In the special case the approach is simplified to demonstrate that some well-known methods can be used for finding exact solutions of nonlinear differential equations. The first integral of the Duffing–Van der Pol equation is found and the general solution of the equation is given in the special case for parameters of the equation. We also demonstrate the efficiency of the method for finding the first integral and the general solution for one of nonlinear second-order ordinary differential equations.

Keywords: Duffing–Van der Pol oscillator, Painlevé test, exact solution, truncated expansion, singular manifold, general solution

Funding Agency Grant Number
Russian Science Foundation 18-11-00209
This research was supported by Russian Science Foundation Grant No. 18-11-00209 "Development of methods for investigation of nonlinear mathematical models".


DOI: https://doi.org/10.1134/S156035471804007X

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Bibliographic databases:

MSC: 34M25
Received: 01.05.2018
Accepted:04.06.2018
Language:

Citation: Nikolay A. Kudryashov, “Exact Solutions and Integrability of the Duffing–Van der Pol Equation”, Regul. Chaotic Dyn., 23:4 (2018), 471–479

Citation in format AMSBIB
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\by Nikolay A. Kudryashov
\paper Exact Solutions and Integrability of the Duffing–Van der Pol Equation
\jour Regul. Chaotic Dyn.
\yr 2018
\vol 23
\issue 4
\pages 471--479
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. N. A. Kudryashov, “On Integrability of the FitzHugh – Rinzel Model”, Nelineinaya dinam., 15:1 (2019), 13–19  mathnet  crossref  elib
    2. Nikolay A. Kudryashov, Dariya V. Safonova, Anjan Biswas, “Painlevé Analysis and a Solution to the Traveling Wave Reduction of the Radhakrishnan – Kundu – Lakshmanan Equation”, Regul. Chaotic Dyn., 24:6 (2019), 607–614  mathnet  crossref
    3. W. Liu, Yu. Zhang, A. M. Wazwaz, Q. Zhou, “Analytic study on triple-s, triple-triangle structure interactions for solitons in inhomogeneous multi-mode fiber”, Appl. Math. Comput., 361 (2019), 325–331  crossref  mathscinet  zmath  isi  scopus
    4. N. A. Kudryashov, “First integrals and general solution of the fokas-lenells equation”, Optik, 195 (2019), 163135  crossref  mathscinet  isi  scopus
    5. N. A. Kudryashov, “Solitary and periodic waves of the hierarchy for propagation pulse in optical fiber”, Optik, 194 (2019), 163060  crossref  isi  scopus
    6. N. A. Kudryashov, “Construction of nonlinear differential equations for description of propagation pulses in optical fiber”, Optik, 192 (2019), 162964  crossref  isi  scopus
    7. N. A. Kudryashov, “Traveling wave reduction of the modified KdV hierarchy: the Lax pair and the first integrals”, Commun. Nonlinear Sci. Numer. Simul., 73 (2019), 472–480  crossref  mathscinet  isi  scopus
    8. N. A. Kudryashov, “Lax pair and first integrals of the traveling wave reduction for the KdV hierarchy”, Appl. Math. Comput., 350 (2019), 323–330  crossref  mathscinet  zmath  isi  scopus
    9. N. A. Kudryashov, “Traveling wave solutions of the generalized nonlinear Schrödinger equation with cubic-quintic nonlinearity”, Optik, 188 (2019), 27–35  crossref  isi  scopus
    10. N. A. Kudryashov, “General solution of traveling wave reduction for the Kundu–Mukherjee–Naskar model”, Optik, 186 (2019), 22–27  crossref  isi  scopus
    11. N. A. Kudryashov, “General solution of the traveling wave reduction for the perturbed Chen–Lee–Liu equation”, Optik, 186 (2019), 339–349  crossref  isi  scopus
    12. N. A. Kudryashov, “First integrals and general solution of the traveling wave reduction for Schrödinger equation with anti-cubic nonlinearity”, Optik, 185 (2019), 665–671  crossref  isi  scopus
    13. N. A. Kudryashov, “A generalized model for description of propagation pulses in optical fiber”, Optik, 189 (2019), 42–52  crossref  isi  scopus
    14. N. A. Kudryashov, “The first integrals and exact solutions of a two-component Belousov–Zhabotinskii reaction system”, VII International Conference Problems of Mathematical Physics and Mathematical Modelling, Journal of Physics Conference Series, 1205, IOP Publishing Ltd, 2019, 012030  crossref  isi  scopus
    15. N. A. Kudryashov, “First integrals and solutions of the traveling wave reduction for the Triki–Biswas equation”, Optik, 185 (2019), 275–281  crossref  isi  scopus
    16. N. A. Kudryashov, “Exact solutions of the equation for surface waves in a convecting fluid”, Appl. Math. Comput., 344 (2019), 97–106  crossref  mathscinet  zmath  isi  scopus
    17. N. A. Kudryashov, “On general solutions of two nonlinear ordinary differential equations”, International Conference on Numerical Analysis and Applied Mathematics (ICNAAM-2018), AIP Conf. Proc., 2116, eds. T. Simos, C. Tsitouras, Amer. Inst. Phys., 2019, 270002  crossref  mathscinet  isi  scopus
    18. N. A. Kudryashov, “Remarks on the Fuchs indices and the first integrals for nonlinear ordinary differential equations”, VII International Conference Problems of Mathematical Physics and Mathematical Modelling, Journal of Physics Conference Series, 1205, IOP Publishing Ltd, 2019, 012031  crossref  mathscinet  isi  scopus
    19. Oswaldo González-Gaxiola, Anjan Biswas, Mir Asma, Abdullah Kamis Alzahrani, “Optical Dromions and Domain Walls with the Kundu – Mukherjee – Naskar Equation by the Laplace – Adomian Decomposition Scheme”, Regul. Chaotic Dyn., 25:4 (2020), 338–348  mathnet  crossref
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