Zero-Hopf bifurcation in the FitzHugh-Nagumo system

Abstract

We characterize the values of the parameters for which a zero--Hopf equilibrium point takes place at the singular points, namely, $O$ (the origin), $P_{+}$ and $P_{-}$ in the FitzHugh-Nagumo system. Thus we find two $2$--parameter families of the FitzHugh-Nagumo system for which the equilibrium point at the origin is a zero-Hopf equilibrium. For these two families we prove the existence of a periodic orbit bifurcating from the zero--Hopf equilibrium point $O$. We prove that exist three $2$--parameter families of the FitzHugh-Nagumo system for which the equilibrium point at $P_{+}$ and $P_{-}$ is a zero-Hopf equilibrium point. For one of these families we prove the existence of $1$, or $2$, or $3$ periodic orbits borning at $P_{+}$ and $P_{-}$.

References in corpus (2)

• Homoclinic Orbits of the FitzHugh-Nagumo Equation: Bifurcations in the Full System
• Homoclinic Orbits of the FitzHugh-Nagumo Equation: The Singular-Limit