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master's thesis
Monotona dinamika Frenkel-Kontorovina modela i primjene
Zagreb: University of Zagreb, Faculty of Science, 2016. urn:nbn:hr:217:612658
Gregurić, Tena
University of Zagreb
Faculty of Science
Department of Mathematics

Institutional repository: Repository of the Faculty of Science

Cite this document

Gregurić, T. (2016). Monotona dinamika Frenkel-Kontorovina modela i primjene (Master's thesis). Zagreb: University of Zagreb, Faculty of Science. Retrieved from https://urn.nsk.hr/urn:nbn:hr:217:612658

Gregurić, Tena. "Monotona dinamika Frenkel-Kontorovina modela i primjene." Master's thesis, University of Zagreb, Faculty of Science, 2016. https://urn.nsk.hr/urn:nbn:hr:217:612658

Gregurić, Tena. "Monotona dinamika Frenkel-Kontorovina modela i primjene." Master's thesis, University of Zagreb, Faculty of Science, 2016. https://urn.nsk.hr/urn:nbn:hr:217:612658

Gregurić, T. (2016). 'Monotona dinamika Frenkel-Kontorovina modela i primjene', Master's thesis, University of Zagreb, Faculty of Science, accessed 22 March 2025, https://urn.nsk.hr/urn:nbn:hr:217:612658

Gregurić T. Monotona dinamika Frenkel-Kontorovina modela i primjene [Master's thesis]. Zagreb: University of Zagreb, Faculty of Science; 2016 [cited 2025 March 22] Available at: https://urn.nsk.hr/urn:nbn:hr:217:612658

T. Gregurić, "Monotona dinamika Frenkel-Kontorovina modela i primjene", Master's thesis, University of Zagreb, Faculty of Science, Zagreb, 2016. Available at: https://urn.nsk.hr/urn:nbn:hr:217:612658

Metadata
TitleMonotona dinamika Frenkel-Kontorovina modela i primjene
AuthorTena Gregurić
MentorSiniša Slijepčević (mentor)
Committee memberSiniša Slijepčević (predsjednik povjerenstva)
Committee memberPavle Pandžić (član povjerenstva)
Committee memberAndrej Dujella (član povjerenstva)
Committee memberJosip Tambača (član povjerenstva)
GranterUniversity of Zagreb
Faculty of Science
(Department of Mathematics)
Zagreb
Defense date and country2016-09-23, Croatia
Scientific / art field,
discipline and subdiscipline		NATURAL SCIENCES
Mathematics
Abstract
U ovom radu bavili smo se svojstvima monotonih dinamičkih sustava te smo se posebno fokusirali na Frenkel-Kontorovin model. Dinamiku nekog modela opisujemo sustavom običnih diferencijalnih jednadžbi. Monoton dinamički sustav je dinamički sustav na metričkom prostoru koji čuva uređaj, odnosno, ako početna stanja zadovoljavaju neki uređaj, onda i sva buduća stanja zadovoljavaju taj uređaj. Kažemo da sustav običnih diferencijalnih jednadžbi generira polutok. Proučavali smo uvjete konvergencije... More tako definiranog polutoka i svojstva omega graničnog skupa. Nadalje, definirali smo kooperativne i kompetitivne diferencijalne jednadžbe. Kažemo da je sustav običnih diferencijalnih jednadžbi kooperativan ako generira monoton polutok unaprijed (u smjeru budućeg vremena), odnosno, da je kompetitavan ako generira monoton tog unazad (u smjeru prošlog vremena). U zadnjem poglavlju fokusirali smo se na glavnu temu ovog rada, Frenkel-Kontorovin model. To je model koji opisuje dinamiku lanca čestica. Pri tome postoji međudjelovanje svake čestice sa najbližim susjedom u prisutnosti vanjskog periodičnog potencijala. Dodatno, na taj model djelovali smo vanjskom silom F. Zaključili smo da za F=0 uvijek postoji ravnotežna točka. S druge strane, sigurno možemo naći dovoljno veliku silu F takvu da neće postojati niti jedna točka ravnoteže. Frenkel-Kontorovin model ima mnoge praktične primjene i u ovom smo radu naveli dvije od njih. Less
Abstract (english)
In this paper we discussed properties of monotone dynamical systems with a special focus on the Frenkel-Kontorova Model. We modeled dynamics of a model by the system of ordinary differential equations. A monotone dynamical system is a dynamical system on a metric space which has a property that ordered initial states lead to ordered subsequent states. We say that a system of ordinary differential equations generates the semiflow. We studied convergence criteria of the semiflow and some... More properties of the omega limit set. Furthermore, we defined competitive and cooperative differential equations. We say that the system of ordinary differential equations is cooperative if it generates a monotone semiflow in the forward time direction and competitive if it generates monotone semiflow in the backward time direction. In the last chapter we focused on the main topic of this master thesis, the Frenkel-Kontorova Model. That is a model which describes the dynamics of a chain of particles interacting with the nearest neighbors in the presence of an external periodic potential. In addition, we applied an external force to the system. We concluded that when F = 0, there always exists an equilibrium for this model. On the other hand, we also proved that we can find a force F large enough so that there are no equilibria. Practical applications of Frenkel-Kontorova model are numerous and we described two of them in this thesis. Less
Keywords
Keywords (english)
Languagecroatian
URN:NBNurn:nbn:hr:217:612658
Study programmeTitle: Finance and Business Mathematics
Study programme type: university
Study level: graduate
Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
Type of resourceText
File originBorn digital
Access conditionsOpen access
Terms of useLicence In copyright icon
Created on2019-01-25 11:02:01
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