From Puiseux series to invariant algebraic curves: the FitzHugh-Nagumo model

A relationship between Puiseux series satisfying an ordinary differential equation corresponding to a polynomial dynamical system and degrees of irreducible invariant algebraic curves is studied. A bound on the degrees of irreducible invariant algebraic curves for a wide class of polynomial dynamical systems is obtained. It is demonstrated that the Puiseux series near infinity can be used to find irreducible algebraic curves explicitly. As an example, all irreducible invariant algebraic curves for the famous FitzHugh-Nagumo system are obtained.