Invariant algebraic curves for Li\'{e}nard dynamical systems revisited

TL;DR

This paper introduces a new algebraic method to find invariant algebraic curves in polynomial vector fields and applies it to Lie9nard systems, revealing more complex invariant structures than previously known.

Contribution

A novel algebraic approach for identifying invariant algebraic curves in polynomial vector fields and detailed analysis of Lie9nard systems' invariant structures.

Findings

Existence of more complex invariant algebraic curves in Lie9nard systems.

Complete characterization of invariant curves for degree 3, degree 2 Lie9nard systems.

Introduction of an algebraic method applicable to polynomial vector fields.

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.