Invariant algebraic curves for Liénard dynamical systems revisited

Abstract

A novel algebraic method for finding invariant algebraic curves for a polynomial vector field in ${\mathbb{C}}^2$ is introduced. The structure of irreducible invariant algebraic curves for Liénard dynamical systems $x_t=y$, $y_t=-g(x)y-f(x)$ with $\text{deg}f=\text{deg}g+1$ is obtained. It is shown that there exist Liénard systems that possess more complicated invariant algebraic curves than it was supposed before. As an example, all irreducible invariant algebraic curves for the Liénard differential system with $\text{deg}f=3$, $\text{deg}g=2$ are obtained. All these results seem to be new.