A family of approximate inertial manifolds for a Van der Pol/FitzHugh–Nagumo perturbation problem

Abstract

We investigate the stability of a family of approximate inertial manifolds (AIMs) obtained from an ODE containing a perturbation parameter. For two choices of the parameter, the dynamics associated with the equations are already well known: in one case, we have a Van der Pol equation, while in the other setting we obtain a FitzHugh–Nagumo equation. Recently, it has been shown that (a modified version of) each equation admits a sequence of AIMs which converges to the inertial manifold. We show that our model admits a family of AIMs depending on the perturbation parameter. We then investigate the stability of the family of AIMs as the perturbation parameter approaches two different vanishing coefficient limits. These results are intended to shed insight into the continuity properties of inertial manifolds.

Keywords:

• approximate inertial manifolds
• stability theory
• vanishing coefficients
• Van der Pol
• FitzHugh–Nagumo

2000 AMS Subject Classifications :

• primary: 34D20
• secondary: 37C70
• 37D10
• 65L07