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

TL;DR

This paper explores how Puiseux series can be used to find invariant algebraic curves in polynomial dynamical systems, providing bounds on their degrees and explicitly determining all such curves for the FitzHugh-Nagumo model.

Contribution

It establishes a connection between Puiseux series and invariant algebraic curves, deriving degree bounds and explicitly computing these curves for a key biological model.

Findings

Derived degree bounds for invariant algebraic curves in polynomial systems.

Showed how Puiseux series near infinity aid in explicitly finding algebraic curves.

Computed all invariant algebraic curves for the FitzHugh-Nagumo system.

Abstract

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.