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Stability in Hamiltonian Systems: KAM stability versus instability around an invariant torus
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
2017 (English)Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesisAlternative title
Stabilitet inom Hamiltonska System : KAM stabilitet kontra instabilitet kring en invariant torus (Swedish)
Abstract [en]

In his ICM-54 lecture, Kolmogorov introduced a now fundamental result regarding the persistence of a large (in the measure theoretic sense) set of invariant tori, in a certain category of almost-integrable Hamiltonian systems. 44 years later, in his ICM-98 talk, Herman conjectured that given any analytic Hamiltonian system with an invariant diophantine torus, this torus will always be accumulated by a positive measure set of invariant KAM tori, i.e. it will be KAM stable.

In this thesis, we build upon recent results and provide a counterexample in three degrees of freedom to KAM stability around an invariant torus, in the category of smooth Hamiltonian systems. The thesis is self-contained in the sense that it also includes a brief introduction to Hamiltonian systems, as well as an exposition of Kolmogorov's classic result.

Abstract [sv]

Under sin ICM-54 föreläsning introducerade Kolmogorov ett numera fundamentalt resultat angående bevarandet av en måtteoretiskt stor mängd invarianta torusar, inom en viss kategori av nästan intagrabla Hamiltonska system. 44 år senare, under sitt ICM-98 tal, formulerade Herman en förmodan om att en invariant diofantisk torus tillhörande en analytisk Hamiltonian alltid omges av en mängd invarianta KAM torusar av positivt mått.

Detta examensarbete bygger vidare på befintliga resultat och ger i fallet tre frihetsgrader ett motexempel till KAM stabilitet kring en invariant torus, i kategorin glatta Hamiltonska system. Arbetet är självtillräckligt i den mening att det även ges en kort introduktion till Hamiltonska system, samt en exposition av Kolmogorovs klassiska resultat.

Place, publisher, year, edition, pages
2017.
Series
TRITA-MAT-E ; 2017:03
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-201616OAI: oai:DiVA.org:kth-201616DiVA, id: diva2:1074362
Subject / course
Mathematics
Educational program
Master of Science - Mathematics
Supervisors
Examiners
Available from: 2017-02-15 Created: 2017-02-15 Last updated: 2017-02-17Bibliographically approved

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