## Numerical KAM theory and backward error analysis for symplectic methods applied to (quasi-)periodically perturbed Hamiltonian ODE

##### Summary

Recenly, a model for tidal waves in shallow areas has been reconsidered, previously studied in the 1990's. The goal was to study mixing and transport due to chaotic motion in the periodic Poincar´e map. In this thesis, the study of this system is continued, shifting, however, the focus to regular instead of chaotic motion. Regular motion has been studied using Liouville (complete) integrability, action-angle coordinates and KAM theorems (we do not consider Nekhoroshev type theorems).
The Hamiltonian of the unperturbed tidal wave system is planar, so Liouville integrable and a first result of this thesis
is the explicit form of the action angle coordinates of the unperturbed tidal wave system.
Furthermore, from a mathematical viewpoint the tidal wave system is also interesting, as it is the natural
third example in the sequence: harmonic oscillator, pendulum , tidal wave system. Therefore it presents an interesting problem to test KAM theory.
The perturbed tidal wave system is a periodically perturbed planar Hamiltonian system (1+ 1/2 d.o.f.). A second result is a proof of existence of persistent invariant tori in the periodic Poincar´e map using a (already developed) KAM theorem for (quasi)-periodically perturbed systems.
In this thesis we are interested in the numerical side: A third result is the proof of a "numerical” KAM theory for numerically integrated periodically perturbed systems, where the numerical integration is done using a symplectic integrator. This result is based on non-autonomous backward error analysis i.e. interpolation of symplectic maps by Hamiltonian flows. This numerical KAM theorem is not able to prove the existence of invariant tori of the periodic Poincar´e map of the symplectically integrated the tidal wave system. Therefore we present an ‘approximate’ KAM theorem, which proves, up to an assumption,
the existence of ‘approximate’ KAM tori in the numerical, periodic Poincar´e map over exponentially long
times, which is the final result of this thesis.
Finally, we mention that backward error analysis (BEA) is of central importance to this thesis. Indeed in
both the theoretical system as well as the numerically integrated system, the proof of the above mentioned
KAM theorems was through the continuous setting (Kolmogorov/Arnolds setting i.e. flows) and not the
discrete (Mosers setting i.e. symplectic maps) and to this end BEA was used to embed the discrete
symplectic integrator into a Hamiltonian flow. Since BEA explains the well-behavedness of symplectic
integrators applied to autonomous Hamiltonian problems a minor part of this thesis was devoted to the
development of modified equation analysis (a type of BEA) for non-autonomous ODE.