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Computational dynamics – real and complex
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
2017 (English)Doctoral thesis, comprehensive summary (Other academic)
Description
Abstract [en]

The PhD thesis considers four topics in dynamical systems and is based on one paper and three manuscripts.

In Paper I we apply methods of interval analysis in order to compute the rigorous enclosure of rotation number. The described algorithm is supplemented with a method of proving the existence of periodic points which is used to check rationality of the rotation number.

In Manuscript II we provide a numerical algorithm for computing critical points of the multiplier map for the quadratic family (i.e., points where the derivative of the multiplier with respect to the complex parameter vanishes).

Manuscript III concerns continued fractions of quadratic irrationals. We show that the generating function corresponding to the sequence of denominators of the best rational approximants of a quadratic irrational is a rational function with integer coefficients. As a corollary we can compute the Lévy constant of any quadratic irrational explicitly in terms of its partial quotients.

Finally, in Manuscript IV we develop a method for finding rigorous enclosures of all odd periodic solutions of the stationary Kuramoto-Sivashinsky equation. The problem is reduced to a bounded, finite-dimensional constraint satisfaction problem whose solution gives the desired information about the original problem. Developed approach allows us to exclude the regions in L2, where no solution can exist.

Place, publisher, year, edition, pages
Uppsala: Department of Mathematics, Uppsala University , 2017. , p. 26
Series
Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 103
Keywords [en]
Continued fractions, Generating functions, Rotation numbers, Rigorous computations, Interval analysis, Interval arithmetic, Multipliers, Quadratic map, Kuramoto-Sivashinsky equation
National Category
Mathematics
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:uu:diva-332280ISBN: 978-91-506-2665-0 (print)OAI: oai:DiVA.org:uu-332280DiVA, id: diva2:1152860
Public defence
2017-12-15, 4101, Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, 09:00 (English)
Opponent
Supervisors
Available from: 2017-11-23 Created: 2017-10-26 Last updated: 2017-11-23
List of papers
1. Rigorous enclosures of rotation numbers by interval methods.
Open this publication in new window or tab >>Rigorous enclosures of rotation numbers by interval methods.
2016 (English)In: Journal of Computational Dynamics, ISSN 2158-2491, Vol. 3, no 1, p. 81-91Article in journal (Refereed) Published
Abstract [en]

We apply set-valued numerical methods to compute an accurate enclosure of the rotation number. The described algorithm is supplemented with a method of proving the existence of periodic points, which is used to check the rationality of the rotation number. A few numerical experiments are presented to show that the implementation of interval methods produces a good enclosure of the rotation number of a circle map.

Keywords
rotation number, circle map, rigorous computation, interval arithmetic, set-valued mathematics
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-332153 (URN)10.3934/jcd.2016004 (DOI)
Available from: 2017-10-24 Created: 2017-10-24 Last updated: 2017-10-26
2. Critical points of the multiplier map for the quadratic family.
Open this publication in new window or tab >>Critical points of the multiplier map for the quadratic family.
(English)Manuscript (preprint) (Other academic)
Keywords
multipliers, quadratic family
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-332157 (URN)
Available from: 2017-10-24 Created: 2017-10-24 Last updated: 2017-10-26
3. Quadratic irrationals, generating functions and Lévy constants.
Open this publication in new window or tab >>Quadratic irrationals, generating functions and Lévy constants.
(English)Manuscript (preprint) (Other academic)
Keywords
continued fractions, generating functions
National Category
Mathematical Analysis
Identifiers
urn:nbn:se:uu:diva-332158 (URN)
Available from: 2017-10-24 Created: 2017-10-24 Last updated: 2017-10-26
4. Finding all solutions of stationary Kuramoto-Sivashinsky equations.
Open this publication in new window or tab >>Finding all solutions of stationary Kuramoto-Sivashinsky equations.
(English)Manuscript (preprint) (Other academic)
National Category
Mathematics
Identifiers
urn:nbn:se:uu:diva-332279 (URN)
Available from: 2017-10-26 Created: 2017-10-26 Last updated: 2017-10-26

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Output format
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