  It is known that the Swift-Hohenberg equation $\partial u/\partial t =
-(\partial_x^2 + 1)^2u + \varepsilon (u-u^3)$ can be reduced to the
Ginzburg-Landau equation (amplitude equation) $\partial A/\partial t =
4\partial_x^2 A + \varepsilon (A-3A|A|^2)$ by means of the singular
perturbation method. This means that if $\varepsilon >0$ is sufficiently small,
a solution of the latter equation provides an approximate solution of the
former one. In this paper, a reduction of a certain class of a system of
nonlinear parabolic equations $\partial u/\partial t = \mathcal{P}u +
\varepsilon f(u)$ is proposed. An amplitude equation of the system is defined
and an error estimate of solutions is given. Further, it is proved under
certain assumptions that if the amplitude equation has a stable steady state,
then a given equation has a stable periodic solution . In particular, near the
periodic solution, the error estimate of solutions holds uniformly in $t>0$.
