Hopf Bifurcation in an Oscillatory-Excitatory Reaction-Diffusion Model with Spatial heterogeneity

TL;DR

This paper analyzes a reaction-diffusion model with spatial heterogeneity, demonstrating Hopf bifurcation and oscillatory behavior, supported by theoretical proofs and numerical simulations.

Contribution

It introduces a generalized FitzHugh-Nagumo model with space-dependent excitability, proving Hopf bifurcation and deriving the center-manifold equation.

Findings

Existence of Hopf bifurcation in the model

Coexistence of stationary and oscillatory phenomena

Numerical simulations confirm theoretical results

Abstract

We focus on the qualitative analysis of a reaction-diffusion with spatial heterogeneity. The system is a generalization of the well known FitzHugh-Nagumo system in which the excitability parameter is space dependent. This heterogeneity allows to exhibit concomitant stationary and oscillatory phenomena. We prove the existence of an Hopf bifurcation and determine an equation of the center-manifold in which the solution asymptotically evolves. Numerical simulations illustrate the phenomenon.