Invariant algebraic curves for Li\'{e}nard 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 Li\'{e}nard 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 Li\'{e}nard systems that possess more complicated invariant algebraic curves than it was supposed before. As an example, all irreducible invariant algebraic curves for the Li\'{e}nard differential system with $\text{deg} f=3$, $\text{deg} g=2$ are obtained. All these results seem to be new.