On the transition between autonomous and nonautonomous systems: the case of FitzHugh-Nagumo's model

TL;DR

This paper investigates the transition from autonomous to nonautonomous dynamics in the FitzHugh-Nagumo model, revealing a two-step nonautonomous bifurcation involving the collapse of an integral manifold and the emergence of a globally attracting solution.

Contribution

It introduces a parametric interpolation framework for FitzHugh-Nagumo models, demonstrating a novel two-step nonautonomous bifurcation pattern with rigorous analysis and validation.

Findings

Identification of a two-step nonautonomous bifurcation process.

Construction of a family of systems with an attracting integral manifold.

Rigorous validation of bifurcation phenomena in periodically forced cases.

Abstract

This work deals with a parametric linear interpolation between an autonomous FitzHugh-Nagumo model and a nonautonomous skewed-problem with the same fundamental structure. This paradigmatic example allows to construct a family of nonautonomous dynamical systems with an attracting integral manifold and a hyperbolic repelling trajectory located within the nonautonomous set enclosed by the integral manifold. Upon the variation of the parameter the integral manifold collapses, the hyperbolic repelling solution disappears and a unique globally attracting hyperbolic solution arises in what could be considered yet another nonautonomous Hopf bifurcation pattern. Interestingly, the three phenomena do not happen at the same critical value of the parameter, yielding thus an example of a nonautonomous bifurcation in two steps. We provide a mathematical description of the dynamical objects at play and analyze the periodically forced case via rigorously validated continuation.