Unfolding a Hopf bifurcation in a linear reaction-diffusion equation with strongly localized impurity existence of breathing pulses

TL;DR

This paper develops a framework to analyze the emergence of breathing pulses in reaction-diffusion systems near a Hopf bifurcation, emphasizing the roles of linearity and nonlinearity in fast variables.

Contribution

It introduces a novel approach combining singular perturbation, normal form, and center manifold theories to explicitly derive stationary pulses and analyze their stability near Hopf bifurcations.

Findings

Explicit expressions for stationary pulses derived

Verification of theoretical results through numerical simulation

Insights into how fast variable nonlinearities influence breathing pulses

Abstract

This paper presents a general framework to derive the weakly nonlinear stability near a Hopf bifurcation in a special class of multi-scale reaction-diffusion equations. The main focus is on how the linearity and nonlinearity of the fast variables in system influence the emergence of the breathing pulses when the slow variables are linear and the bifurcation parameter is around the Hopf bifurcation point. By applying the matching principle to the fast and slow changing quantities and using the relevant theory of singular perturbation, we obtain explicit expressions for the stationary pulses. Then, the normal form theory and the center manifold theory are applied to give Hopf normal form expressions. Finally, one of these expressions is verified by the numerical simulation.