Integrability and Jacobi Last Multipliers of Cubic Li'{e}nard Differential Equations with Quadratic Damping
DOI:
https://doi.org/10.5890/DNC.2020.12.002Abstract
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.References
[1] Singer, M.F. (1992), Liouvillian first integrals of differential systems, Trans. Amer. Math. Soc., 333, 673-688.
[2] Christopher, C. (1994), Invariant algebraic curves and conditions for a centre, Proc. Roy. Soc. Edinburgh Sect. A, 124(6), 1209-1229.
[3] Gin{e}, J. and Valls, C. (2019), Liouvillian integrability of a general Rayleigh-Duffing oscillator, J. Nonlin. Math. Phys., 26, 169-187.
[4] Demina, M.V. (2018), Novel algebraic aspects of {L}iouvillian integrability for two-dimensional polynomial dynamical systems, Phys. Lett. A, 382(20), 1353-1360.
[5] Demina, M.V. and Sinelshchikov, D.I. (2019), Integrability Properties of Cubic Li{e}nard Oscillators with Linear Damping, Symmetry, 11(11), 1378.
[6] Nucci, M.C. and Leach, P.G.L. (2009), An old method of Jacobi to find Lagrangians, J. Nonlinear Math. Phys., 16(4), 431-441. %doi: 10.1142/S14029251090 0 0467
[7] D'Ambrosi, G. and Nucci, M.C. (2009), Lagrangians for equations of Painlev{e} type by means of the Jacobi last multiplier. J. Nonlinear Math. Phys., 16, 61-71.
[8] Demina, M.V. (2018), Invariant algebraic curves for {L}i{e}nard dynamical systems revisited, Appl. Math. Lett., 84, 42-48.
[9] Stachowiak, T. (2019), Hypergeometric first integrals of the Duffing and van der Pol oscillators, ph{Journal of Differential Equations}, 266(9), 5895-5911.
[10] Demina, M.V. (2018), Invariant surfaces and {D}arboux integrability for non-autonomous dynamical systems in the plane. J. Phys. A: Math. Theor., 51, 505202.
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