Resumen de Dynamic Analysis of FitzHugh-Nagumo System

Wenzhe Cui

• In this paper, we prove the existence of traveling train and impulse in FitzHughNagumo system ut = ux x − u(u − 1)(u − a) − v, vt = ε(u − γv) by studying a saddle-node bifurcation and a Bogdanov-Takens bifurcation of the corresponding three-dimensional system x˙ = z, y˙ = b(x − dy), z˙ = x(x − 1)(x − a) + y + cz.

  The bifurcation analysis of the three-dimensional system indicates that the number of steady-state solutions in FitzHugh-Nagumo system can change through saddle-node bifurcation of the three-dimensional system, and there are different parameter values for which the three-dimensional system can have a limit cycle or a homoclinic loop, which implies that FitzHugh-Nagumo system can have a traveling train or an impulse for some specific parameters.