The number of limit cycles of the FitzHugh nerve system

Abstract

In this paper we give a complete analysis of the number of limit cycles of the FitzHugh nerve system. First, we prove the uniqueness of the limit cycle when the unique equilibrium is a source. We then show that the system has two limit cycles if the unique equilibrium is a sink and limit cycles exist. We will also show that the mathematical study of limit cycles for FitzHugh nerve systems is related to Hilbert’s 16 th ^{\mbox {th}} problem and is therefore an important area of study.