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These notes were written during the 9th and 10th sessions of the subjectDynamical Systems II coursed at DTU (Denmark) during the Winter Semester2015-2016, and later extended in February 2017. They aim to provide studentswith a theoretical and numerical background for the computation of periodicorbits using Newton's method. We focus on periodically perturbedquasi-integrable systems (using the forced pendulum as an example) and hence wetake advantage of the Melnikov method to get first guesses. However, these wellknown techniques are general enough to be applied in other type of systems.Periodic orbits are computed by solving a fixed-point equation for thestroboscopic map, which is very fast and precise. We also consider computingthe Poincar\'e map and compare both strategies. In both cases we show how tocompute the Jacobian of the maps, which is necessary for the Newton method, bymeans of variational equations and the implicit function theorem. Someexercises are proposed along the notes, whose solutions can be found ingithub.com/a-granados. The notes themselves do not contain any reference,although everything described here is well known in the Dynamical Systemscommunity. A typical reference for the Melnikov method for subharmonic orbitsis the book [GucHol83]. More about the variational equations and theirnumerical applications can be found in the notes [Sim].

Notes on the computation of periodic orbits using Newton and Melnikov's method