  In the spectral stability analysis of localized patterns to singular
perturbed evolution problems, one often encounters that the Evans function
respects the scale separation. In such cases the Evans function of the full
linear stability problem can be approximated by a product of a slow and a fast
reduced Evans function, which correspond to properly scaled slow and fast
singular limit problems. This feature has been used in several spectral
stability analyses in order to reduce the complexity of the linear stability
problem. In these studies the factorization of the Evans function was
established via geometric arguments that need to be customized for the specific
equations and solutions under consideration. In this paper we develop an
alternative factorization method. In this analytic method we use the Riccati
transformation and exponential dichotomies to separate slow from fast dynamics.
We employ our factorization procedure to study the spectra associated with
spatially periodic pulse solutions to a general class of multi-component,
singularly perturbed reaction-diffusion equations. Eventually, we obtain
expressions of the slow and fast reduced Evans functions, which describe the
spectrum in the singular limit. The spectral stability of localized periodic
patterns has so far only been investigated in specific models such as the
Gierer-Meinhardt equations. Our spectral analysis significantly extends and
formalizes these existing results. Moreover, it leads to explicit instability
criteria.
