---

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

Download Full Text PDF

 

Abstract

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.

Acknowledgments

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

References

1. [1]	Singer, M.F. (1992), Liouvillian first integrals of differential systems, Trans. Amer. Math. Soc., 333, 673-688.

2. [2]	Christopher, C. (1994), Invariant algebraic curves and conditions for a centre, Proc. Roy. Soc. Edinburgh Sect. A, 124(6), 1209-1229.

3. [3]	Gin{e}, J. and Valls, C. (2019), Liouvillian integrability of a general Rayleigh-Duffing oscillator, J. Nonlin. Math. Phys., 26, 169-187.

4. [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. [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. [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. [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. [8]	Demina, M.V. (2018), Invariant algebraic curves for {L}i{e}nard dynamical systems revisited, Appl. Math. Lett., 84, 42-48.

9. [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. [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.