  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.
