The high-order Euler method and the spin-orbit model A fast algorithm for solving differential equations with small, smooth nonlinearity

摘要

We present an algorithm for the rapid numerical integration of smooth,time-periodic differential equations with small nonlinearity, particularlysuited to problems with small dissipation. The emphasis is on speed withoutcompromising accuracy and we envisage applications in problems whereintegration over long time scales is required; for instance, orbit probabilityestimation via Monte Carlo simulation. We demonstrate the effectiveness of ouralgorithm by applying it to the spin-orbit problem, for which we have derivedanalytical results for comparison with those that we obtain numerically. Amongother tests, we carry out a careful comparison of our numerical results withthe analytically predicted set of periodic orbits that exists for givenparameters. Further tests concern the long-term behaviour of solutions movingtowards the quasi-periodic attractor, and capture probabilities for theperiodic attractors computed from the formula of Goldreich and Peale. Weimplement the algorithm in standard double precision arithmetic and show thatthis is adequate to obtain an excellent measure of agreement between analyticalpredictions and the proposed fast algorithm.