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arXiv · 1810.01241

The method of Puiseux series and invariant algebraic curves

Maria V. Demina
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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.

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Maria V. Demina. 2018-09-25. The method of Puiseux series and invariant algebraic curves. https://doi.org/10.1142/s0219199721500073

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