A first order differential equation with a periodic operator coefficient acting in a pair of Hilbert spaces is considered. This setting models both elliptic equations with periodic coefficients in a cylinder and parabolic equations with time periodic coefficients. Our main results are a construction of a pointwise projector and a spectral splitting of the system into a finite-dimensional system of ordinary differential equations with constant coefficients and an infinite-dimensional part whose solutions have better properties in a certain sense. This complements the well-known asymptotic results for periodic hypoelliptic problems in cylinders and for elliptic problems in quasicylinders obtained by P. Kuchment and S. A. Nazarov, respectively. As an application, a center manifold reduction is presented for a class of nonlinear ordinary differential equations in Hilbert spaces with periodic coefficients. This result generalizes the known case with constant coefficients explored by A. Mielke.