Invariant algebraic surfaces of the FitzHugh-Nagumo system

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.