  We present an algorithm for the rapid numerical integration of smooth,
time-periodic differential equations with small nonlinearity, particularly
suited to problems with small dissipation. The emphasis is on speed without
compromising accuracy and we envisage applications in problems where
integration over long time scales is required; for instance, orbit probability
estimation via Monte Carlo simulation. We demonstrate the effectiveness of our
algorithm by applying it to the spin-orbit problem, for which we have derived
analytical results for comparison with those that we obtain numerically. Among
other tests, we carry out a careful comparison of our numerical results with
the analytically predicted set of periodic orbits that exists for given
parameters. Further tests concern the long-term behaviour of solutions moving
towards the quasi-periodic attractor, and capture probabilities for the
periodic attractors computed from the formula of Goldreich and Peale. We
implement the algorithm in standard double precision arithmetic and show that
this is adequate to obtain an excellent measure of agreement between analytical
predictions and the proposed fast algorithm.
