Stabilnost diferencijalnih jednadžbi

master's thesis

• Author: Ramona Mihaela Vopel
• Mentor: Lavoslav Čaklović (mentor)
• Committee member: Lavoslav Čaklović (predsjednik povjerenstva)
• Committee member: Dražen Adamović (član povjerenstva)
• Committee member: Pavle Pandžić (član povjerenstva)
• Committee member: Mirko Primc (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2016-09-27, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Diferencijalne jednadžbe imaju važnu ulogu u mnogim granama znanosti i industrije. Predmet proučavanja teorije stabilnosti je asimptotsko ponašanje funkcija u okviru familije rješenja, tj. da li se one približavaju ili razilaze jedna od druge na beskonačnom intervalu. U ovom radu smo se usredotočili na obične diferencijalne jednadžbe, njihovu podjelu, te osnovne teoreme o postojanju i jedinstvenosti rješenja. Dana je kratka povijest razvoja teorije stabilnosti , kao i različite definicije stabilnosti. Pod određenim uvjetima, pitanje koje teorija stabilnosti postavlja se može svesti na problem koji razmatra svojstvene vrijednosti matrica. U analizi stabilnosti rješenja običnih diferencijalnih jednadžbi smo koristili dvije metode, direktnu metodu te metodu linearizacije. Također smo pokazali neke od metoda određivanja Lyapunovih funkcija.

Abstract (english)

Differential equations have come to play an important role in many branches of science and industry. Stability theory addresses the following questions: Will a nearby orbit indefinitely stay close to a given orbit? Will it converge to the given orbit? We describe the ordinary differential equations, their classification, as well as the basic theorems of existence and uniqueness of solutions. A brief history of stability theory development is given, as well as stability definitions. Various criteria have been developed to prove stability or instability of an orbit. Under favorable circumstances, above question may be reduced to a well-studied problem involving eigenvalues of matrices. In stability analysis of solutions of ODE, two methods are used: direct method and method of linearisation. Some techniques are shown to determine the Lyapunov function.

• Keywords: obične diferencijalne jednadžbe; teorija stabilnosti; direktna metoda; metoda linearizacije; Lyapunove funkcije
• Keywords (english): ordinary differential equations; stability theory; direct method; method of linearisation; Lyapunov function
• Language: croatian
• URN:NBN: urn:nbn:hr:217:572206
• 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: 2019-02-19 15:19:11