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 Liénard 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 Liénard differential equations with quadratic damping do not have algebraic limit cycles.