Computing periodic orbits of the three-body problem: Effective convergence of Newton’s method on the surface of section

We consider Newton’s method for computing periodic orbits of dynamical systems as fixed points on a surface of section and seek to clarify and evaluate the method’s uncertainty of convergence. Several fixed points of various multiplicities, both stable and unstable are computed in a new version of Hill’s problem. Newton’s method is applied with starting points chosen randomly inside the maximum possible—for any method—circle of convergence. The employment of random starting points is continued until one of them leads to convergence, and the process is repeated a thousand times for each fixed point. The results show that on average convergence occurs with very few starting points and non-converging iterations being wasted.