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

TL;DR

This paper discusses methods for computing periodic orbits in dynamical systems using Newton's method combined with Melnikov's approach for initial guesses, emphasizing efficiency and applicability to perturbed quasi-integrable systems.

Contribution

It provides a detailed theoretical and numerical framework for computing periodic orbits using Newton's method, Melnikov's method for initial guesses, and compares strategies involving the stroboscopic and Poincaré maps.

Findings

Fast and precise computation of periodic orbits via fixed-point equations

Comparison of stroboscopic map and Poincaré map strategies

Explicit computation of Jacobians using variational equations

Abstract

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].