  This paper investigates the asymptotic behaviour of solutions of periodic
evolution equations. Starting with a general result concerning the quantified
asymptotic behaviour of periodic evolution families we go on to consider a
special class of dissipative systems arising naturally in applications. For
this class of systems we analyse in detail the spectral properties of the
associated monodromy operator, showing in particular that it is a so-called
Ritt operator under a natural 'resonance' condition. This allows us to deduce
from our general result a precise description of the asymptotic behaviour of
the corresponding solutions. In particular, we present conditions for rational
rates of convergence to periodic solutions in the case where the convergence
fails to be uniformly exponential. We illustrate our general results by
applying them to concrete problems including the one-dimensional wave equation
with periodic damping.
