  These notes were written during the 9th and 10th sessions of the subject
Dynamical Systems II coursed at DTU (Denmark) during the Winter Semester
2015-2016, and later extended in February 2017. They aim to provide students
with a theoretical and numerical background for the computation of periodic
orbits using Newton's method. We focus on periodically perturbed
quasi-integrable systems (using the forced pendulum as an example) and hence we
take advantage of the Melnikov method to get first guesses. However, these well
known techniques are general enough to be applied in other type of systems.
Periodic orbits are computed by solving a fixed-point equation for the
stroboscopic map, which is very fast and precise. We also consider computing
the Poincar\'e map and compare both strategies. In both cases we show how to
compute the Jacobian of the maps, which is necessary for the Newton method, by
means of variational equations and the implicit function theorem. Some
exercises are proposed along the notes, whose solutions can be found in
github.com/a-granados. The notes themselves do not contain any reference,
although everything described here is well known in the Dynamical Systems
community. A typical reference for the Melnikov method for subharmonic orbits
is the book [GucHol83]. More about the variational equations and their
numerical applications can be found in the notes [Sim].
