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Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA


Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania


Integrability and Jacobi Last Multipliers of Cubic Li'{e}nard Differential Equations with Quadratic Damping

Discontinuity, Nonlinearity, and Complexity 9(4) (2020) 499--507 | DOI:10.5890/DNC.2020.12.002

Maria V. Demina

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

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We solve completely the problem of Liouvillian integrability for cubic Li\'{e}nard differential equations with quadratic damping. %Our results are applicable for a wide family of dynamical systems. Our main tool is the method of Puiseux series. We find necessary and sufficient conditions for equations under consideration to have Jacobi last multipliers of a special form. It turns out that some particular sub--families being Liouvillian non--integrable possess Jacobi last multipliers. The Jacobi last multipliers give rise to non--standard Lagrangians and it is an interesting property of these dynamical systems. In addition, we prove that cubic Li\'{e}nard differential equations with quadratic damping do not have algebraic limit cycles.


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


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