The method of Puiseux series and invariant algebraic curves

TL;DR

This paper derives explicit formulas for cofactors of invariant algebraic curves in polynomial dynamical systems, establishes conditions for finiteness of such curves, and analyzes Libe9nard systems to show they can have at most two invariant algebraic curves, implying non-integrability.

Contribution

It provides explicit cofactor expressions, a finiteness criterion for invariant algebraic curves, and characterizes invariant curves in Libe9nard systems, revealing their limited number and non-integrability.

Findings

Explicit cofactor formulas for invariant algebraic curves.

Finiteness condition for the number of invariant algebraic curves.

Libe9nard systems with certain degree relations have at most two invariant algebraic curves.

Abstract

An explicit expression for the cofactor related to an irreducible invariant algebraic curve of a polynomial dynamical system in the plane is derived. A sufficient condition for a polynomial dynamical system in the plane to have a finite number of irreducible invariant algebraic curves is obtained. All these results are applied to Li\'enard dynamical systems $x_t=y$, $y_t=-f(x)y-g(x)$ with $\text{deg}\, f<\text{deg}\,g<2\,\text{deg}\,f+1$. The general structure of their irreducible invariant algebraic curves and cofactors is found. It is shown that Li\'enard dynamical systems with $\text{deg}\, f<\text{deg}\, g<2\,\text{deg}\, f+1$ can have at most two distinct irreducible invariant algebraic curves simultaneously and consequently are not integrable with a rational first integral.