  We consider contractive systems whose trajectories evolve on a compact and
convex state-space. It is well-known that if the time-varying vector field of
the system is periodic then the system admits a unique globally asymptotically
stable periodic solution. Obtaining explicit information on this periodic
solution and its dependence on various parameters is important both
theoretically and in numerous applications. We develop an approach for
approximating such a periodic trajectory using the periodic trajectory of a
simpler system (e.g. an LTI system). Our approximation includes an error bound
that is based on the input-to-state stability property of contractive systems.
We show that in some cases this error bound can be computed explicitly. We also
use the bound to derive a new theoretical result, namely, that a contractive
system with an additive periodic input behaves like a low pass filter. We
demonstrate our results using several examples from systems biology.
