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Computational dynamics – real and complexPrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true});
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PrimeFaces.cw("AccordionPanel","widget_formSmash_responsibleOrgs",{id:"formSmash:responsibleOrgs",widgetVar:"widget_formSmash_responsibleOrgs",multiple:true}); 2017 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Description

##### Abstract [en]

##### Place, publisher, year, edition, pages

Uppsala: Department of Mathematics, Uppsala University , 2017. , p. 26
##### Series

Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 103
##### Keyword [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

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##### Supervisors

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#####

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Available from: 2017-11-23 Created: 2017-10-26 Last updated: 2017-11-23
##### List of papers

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 L^{2}, where no solution can exist.

1. Rigorous enclosures of rotation numbers by interval methods.$(function(){PrimeFaces.cw("OverlayPanel","overlay1152399",{id:"formSmash:j_idt656:0:j_idt663",widgetVar:"overlay1152399",target:"formSmash:j_idt656:0:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

2. Critical points of the multiplier map for the quadratic family.$(function(){PrimeFaces.cw("OverlayPanel","overlay1152402",{id:"formSmash:j_idt656:1:j_idt663",widgetVar:"overlay1152402",target:"formSmash:j_idt656:1:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

3. Quadratic irrationals, generating functions and Lévy constants.$(function(){PrimeFaces.cw("OverlayPanel","overlay1152403",{id:"formSmash:j_idt656:2:j_idt663",widgetVar:"overlay1152403",target:"formSmash:j_idt656:2:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

4. Finding all solutions of stationary Kuramoto-Sivashinsky equations.$(function(){PrimeFaces.cw("OverlayPanel","overlay1152834",{id:"formSmash:j_idt656:3:j_idt663",widgetVar:"overlay1152834",target:"formSmash:j_idt656:3:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

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