Universität zu Köln

Schließauf, Henrik ORCID: 0000-0002-2257-4027 (2022). Twist maps in low regularity and non-periodic settings. PhD thesis, Universität zu Köln.

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Abstract

Symplectic twist maps already appeared in the works of Poincaré. They emerge naturally as discretizations of certain low-dimensional Hamiltonian systems and offer a nice handle for studying their dynamics. The associated theory had experienced a tremendous boost from the discoveries made by Kolmogorow, Arnold and Moser in the 1960's, and by Aubry and Mather in the 1980's. In this thesis, we will substantiate the usefulness of studying twist maps also in non-periodic and low regularity settings, that is in situations where the classical results are not applicable. In particular, we will concentrate on perturbative methods for near-integrable systems. One typical feature of such systems is the existence of ``approximate first integrals'', so called adiabatic invariants, and their presence usually has strong consequences for the dynamics. Here, we derive growth rates for a large class of non-periodic twist maps depending on the regularity assumptions. As an application, the Fermi-Ulam ping-pong is considered, where the possible growth in velocity is linked to the number of bounded derivatives of the forcing function. Moreover, in systems with adiabatic invariants and almost periodic time-dependence the underlying compact structure enables one to use a generalization of Poincaré's recurrence theorem. By harnessing this fact, we prove that in such systems the set of initial condition leading to escaping orbits typically has measure zero. This is again demonstrated using the ping-pong model. Other applications are found in the Littlewood boundedness problem, where we consider a periodically forced piecewise linear oscillator together with its discontinuous limit case, and also a super-linear oscillator with an almost periodic forcing term. These systems are given by differential equations and thus the mentioned results imply also the Poisson stability of almost every solution. Even in the periodic case, these insights represent valuable contributions due to the low regularity assumptions necessary to obtain them.

Item Type:	Thesis (PhD thesis)
Creators:	Creators Email ORCID ORCID Put Code Schließauf, Henrik h.schliessauf@googlemail.com orcid.org/0000-0002-2257-4027 UNSPECIFIED
URN:	urn:nbn:de:hbz:38-644182
Date:	12 December 2022
Place of Publication:	Köln
Language:	English
Faculty:	Faculty of Mathematics and Natural Sciences
Divisions:	Faculty of Mathematics and Natural Sciences > Department of Mathematics and Computer Science > Mathematical Institute
Subjects:	Mathematics
Uncontrolled Keywords:	Keywords Language Twist map, Fermi-Ulam ping-pong, Littlewood boundedness problem, almost periodic function English
Date of oral exam:	7 October 2022
Referee:	Name Academic Title Kunze, Markus Prof. Dr. Marinescu, George Prof. Dr.
Funders:	SFB/TRR 191 ‘Symplectic Structures in Geometry, Algebra and Dynamics’
Refereed:	Yes
URI:	http://kups.ub.uni-koeln.de/id/eprint/64418