**Authors:**
Michele Bartuccelli, Jonathan Deane, Guido Gentile

**Title:** *
The high-order Euler method and the spin-orbit model
*

**Abstract: **
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.

**Keywords:**
Fast numerics for differential equation; spin-orbit model;
periodic attractors; quasi-periodic attractors; series expansion; perturbation theory.

Michele V. Bartuccelli

Department of Mathematics

University of Surrey

Guildford, GU2 7HX, UK

e-mail: gentile@mat.uniroma3.it

Jonathan H.B. Deane

Department of Mathematics

University of Surrey

Guildford, GU2 7HX, UK

e-mail: j.deane@surrey.ac.uk

Guido Gentile

Dipartimento di Matematica e Fisica

Università Roma Tre

Largo San Leonardo Murialdo 1 - 00146 Roma - Italy

e-mail: gentile@mat.uniroma3.it