Monotona dinamika Frenkel-Kontorovina modela i primjene

master's thesis

• Author: Tena Gregurić
• Mentor: Siniša Slijepčević (mentor)
• Committee member: Siniša Slijepčević (predsjednik povjerenstva)
• Committee member: Pavle Pandžić (član povjerenstva)
• Committee member: Andrej Dujella (član povjerenstva)
• Committee member: Josip Tambača (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2016-09-23, 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 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.

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 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.

• Keywords: monotoni dinamički sustavi; Frenkel-Kontorovin model; diferencijalne jednadžbe
• Keywords (english): monotone dynamical systems; Frenkel-Kontorova model; differential equations
• Language: croatian
• URN:NBN: urn:nbn:hr:217:612658
• Study programme: Title: Finance and Business Mathematics; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2019-01-25 11:02:01