Kam theory for periodic orbits

Abstract

This project develops a theoretical framework for the convergence of the Newton method in a system with a periodic orbit, proving its existence with an explicit computed error. We adapt the analytical techniques of KAM theory with a small KAM iterative scheme in the sense of a loss of part of the analytic domain while finding the periodic orbit.

More explicitly, we present a conceptual background with norm definitions, analytic results and non-degeneracy results to avoid problems with resonances and to help formulate an iterative lemma for the Newton method to finally present a Convergence Theorem, proving the existence and uniqueness of a periodic orbit if the initial error is small enough.

Therefore, we obtain a theoretical result to prove the existence of periodic orbits when the initial error is small enough. This setup can be applied to Computer Assisted Proof for the existence of periodic orbits.