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

dc.rights.license | CC-BY-NC-ND | |

dc.contributor.advisor | Frank, Jason | |

dc.contributor.author | Carere, Francesco | |

dc.date.accessioned | 2022-09-09T02:02:53Z | |

dc.date.available | 2022-09-09T02:02:53Z | |

dc.date.issued | 2022 | |

dc.identifier.uri | https://studenttheses.uu.nl/handle/20.500.12932/42551 | |

dc.description.abstract | 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. | |

dc.description.sponsorship | Utrecht University | |

dc.language.iso | EN | |

dc.subject | An "approximate" KAM theorem was extended from autonomously to quasi-periodically perturbed Hamiltonian ODE on which a symplectic integrator was applied. Central in this result is the use of backward error analysis. This result was applied to a tidal wave system | |

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

dc.type.content | Master Thesis | |

dc.rights.accessrights | Open Access | |

dc.subject.keywords | KAM theory, symplectic integration, symplectic method, KAM theory for symplectic methods, backward error analysis, numerical Hamiltonian ODE, non-autonomous Hamiltonian ODE | |

dc.subject.courseuu | Mathematical Sciences | |

dc.thesis.id | 9789 |