Integrability and Jacobi Last Multipliers of Cubic Li’enard Differential Equations with Quadratic Damping - Demina, Maria V. (Discontinuity, Nonlinearity, and Complexity, 2026)
We solve completely the problem of Liouvillian integrability for cubic Li\’enard 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\’enard differential equations with quadratic damping do not have algebraic limit cycles.