Invariant algebraic surfaces of the FitzHugh-Nagumo system

TL;DR

This paper classifies invariant algebraic surfaces of the FitzHugh-Nagumo system, revealing no such surfaces exist within biological parameter ranges, using a novel approach involving an auxiliary system.

Contribution

It introduces a new method based on an auxiliary system to classify invariant algebraic surfaces of the FitzHugh-Nagumo system, overcoming limitations of previous techniques.

Findings

No invariant algebraic surfaces in biological parameter region.

Complete classification of Darboux polynomials for the system.

Use of auxiliary system to analyze invariant surfaces.

Abstract

In this paper, we characterize all the irreducible Darboux polynomials and polynomial first integrals of FitzHugh-Nagumo (F-N) system. The method of the weight homogeneous polynomials and the characteristic curves is widely used to give a complete classification of Darboux polynomials of a system. However, this method does not work for F-N system. Here by considering the Darboux polynomials of an assistant system associated to F-N system, we classified the invariant algebraic surfaces of F-N system. Our results show that there is no invariant algebraic surface of F-N system in the biological parameters region.