{"?xml":{"@version":"1.0"},"edm:RDF":{"@xmlns:dc":"http://purl.org/dc/elements/1.1/","@xmlns:edm":"http://www.europeana.eu/schemas/edm/","@xmlns:wgs84_pos":"http://www.w3.org/2003/01/geo/wgs84_pos","@xmlns:foaf":"http://xmlns.com/foaf/0.1/","@xmlns:rdaGr2":"http://rdvocab.info/ElementsGr2","@xmlns:oai":"http://www.openarchives.org/OAI/2.0/","@xmlns:owl":"http://www.w3.org/2002/07/owl#","@xmlns:rdf":"http://www.w3.org/1999/02/22-rdf-syntax-ns#","@xmlns:ore":"http://www.openarchives.org/ore/terms/","@xmlns:skos":"http://www.w3.org/2004/02/skos/core#","@xmlns:dcterms":"http://purl.org/dc/terms/","edm:WebResource":[{"@rdf:about":"http://www.dlib.si/stream/URN:NBN:SI:DOC-DE2HVRI0/e9d98bccd4630cf597c221-35-e9464-a6c-/PDF","dcterms:extent":"2641 KB"},{"@rdf:about":"http://www.dlib.si/stream/URN:NBN:SI:DOC-DE2HVRI0/2f2c7517-8e88-4146-b357-d7b6864fe6fe/TEXT","dcterms:extent":"246 KB"},{"@rdf:about":"http://www.dlib.si/stream/URN:NBN:SI:DOC-DE2HVRI0/475c29c9-c95f-4164-b60e-dd386a2c3ce9/WEB","dcterms:extent":"0 KB"}],"edm:ProvidedCHO":{"@rdf:about":"URN:NBN:SI:DOC-DE2HVRI0","dcterms:issued":"2023","dc:creator":"Arcet, Barbara","dc:contributor":"Romanovski, Valery","dc:format":{"@xml:lang":"sl","#text":"V, 140 str., 31 cm"},"dc:identifier":["COBISSID:145318659","URN:URN:NBN:SI:doc-DE2HVRI0"],"dc:language":"sl","dc:publisher":{"@xml:lang":"sl","#text":"B. Arcet"},"dc:source":{"@xml:lang":"sl","#text":"visokošolska dela"},"dc:subject":[{"@xml:lang":"sl","#text":"Diferencialne enačbe"},{"@xml:lang":"sl","#text":"Disertacije"},{"@xml:lang":"en","#text":"dissertations"},{"@xml:lang":"en","#text":"Hamiltonian systems"},{"@xml:lang":"sl","#text":"Hamiltonski sistemi"},{"@xml:lang":"en","#text":"integrability"},{"@xml:lang":"sl","#text":"integrabilnost"},{"@xml:lang":"en","#text":"limit cycles"},{"@xml:lang":"sl","#text":"limitni cikli"},{"@xml:lang":"en","#text":"linearizability"},{"@xml:lang":"sl","#text":"linearizabilnost"},{"@xml:lang":"en","#text":"reversibility"},{"@xml:lang":"sl","#text":"reverzibilnost"},{"@xml:lang":"sl","#text":"sistemi navadnih diferencialnih enačb"},{"@xml:lang":"en","#text":"systems of ordinary differential equations"},{"@xml:lang":"sl","#text":"Univerzitetna in visokošolska dela"}],"dc:title":{"@xml:lang":"sl","#text":"Integrabilnost, linearizabilnost in limitni cikli polinomskih sistemov navadnih diferencialnih enačb| doktorska disertacija|"},"dc:description":[{"@xml:lang":"sl","#text":"The main topic of this doctoral disertation is the qualitative analysis of some families of systems of ordinary differential equations (ODE). We study integrability, linearizability and limit cycles in two- and three-dimensional polynomial systems.In the introduction we define fundamental concepts connected to singular points of systems of ODE and behavior of trajectories in their neigbourhoods. We describe some main methods and algorithms of commutative computational algebra which are essential for the performed analysis of the systems.In the second chapter we define two important properties of n-dimensional systems of ODE, namely integrability and linearizability. First, we introduce a method to get conditions for integrability of a system and then we describe some techniques to prove the sufficiency of such conditions. In order to study linearizability we make a quick introduction into theory of normal forms. We explain a method to find conditions for linearizability of a system and prove an important theorem about linearizability of integrable systems. Using the theory mentioned above we study integrability and linearizability of a quadratic three-dimensional system with (1:-1:-1)-resonant singularity at the origin.The third chapter is devoted to planar systems of ODE and their linearizability, which is very closely related to isochronicity. We explain a method to get the conditions for linearizability of a system which works in some cases when we cannot get them from linearizability quantities. We focus on Hamiltonian systems with homogeneous and nonhomogeneous nonlinearities of degree at most seven. In the fourth part we address the center-focus problem for some reversible cubic systems. We analyse three two-dimensional systems which are transformed into one of the canonical forms of cubic systems with the singularity of center or focus type at the origin. We prove that all the obtained systems are Darboux integrable. At last, we study the orbital reversibility of the systems.In the last chapter we concentrate on limit cycles. We recall one of the key phenomena, responsible for emergence of limit cycles, Hopf bifurcation. We describe a method for investigation of the points at the infinity, Poincar\\e compactification and an approach to analyse a neighborhood of degenerate singular points - directional blow up. Then we study the possibilities for existence of limit cycles in a three-dimensional biochemical model and determine the phase portrait in the first quadrant of a two-dimensional reaction model"},{"@xml:lang":"sl","#text":"Krovna tema pričujoče doktorske disertacije je kvalitativna obravnava nekaterih družin navadnih diferencialnih enačb (NDE). Osrednja pozornost je namenjena ravninskim in tridimenzionalnim polinomskim sistemom ter preiskovanju pogojev, pri katerih se sistemi ponašajo s katero od naslovnih lastnosti: integrabilnostjo, linearizabilnostjo ali prisotnostjo limitnih ciklov. Uvodno poglavje je namenjeno definiciji osnovnih pojmov, ki zadevajo singularne točke in njihove okolice v sistemih NDE. Predstavimo nekaj ključnih metod in algoritmov komutativne računske algebre, ki so bistveni pri preiskovanju sistemov v nadaljevanju dela. V drugem poglavju definiramo dve osrednji lastnosti ?$n$?-dimenzionalnih sistemov NDE, integrabilnost in linearizabilnost. Najprej predstavimo metodo, s katero lahko pridobimo pogoje za integrabilnost sistema, nato pa navedemo nekaj načinov za dokaz zadostnosti teh pogojev. Za preučitev linearizabilnosti se dotaknemo teorije normalnih form, predstavimo način za iskanje pogojev za linearizabilnost sistemov in dokažemo izrek, ki povezuje integrabilnost ter linearizabilnost sistemov NDE. Z uporabo omenjene teorije nato preučimo integrabilnost in linearizabilnost kvadrati\\v cnega tridimenzionalnega sistema z ?$(1:-1:-1)$?-resonantno singularnostjo v izhodišču. Tretje poglavje je namenjeno ravninskim sistemom NDE in njihovi linearizabilnosti, ki je tesno povezana z izohronostjo. Predstavimo metodo za pridobivanje pogojev za linearizabilnost, ko le-teh ne moremo pridobiti iz linearizabilnostnih količin, in sicer iskanje polinomske linearizacije ene od enačb sistema. Pri proučevanju linearizabilnosti se osredotočimo na nekatere Hamiltonske sisteme s homogenimi in nehomogenimi nelinearnostmi stopnje kvečjemu sedem. V četrtem delu disertacije se lotimo problema centra in fokusa za nekatere rever-zibilne kubične sisteme. V tem smislu preiskujemo tri sisteme, ki so z ustrezno transformacijo prevedeni v eno izmed kanoničnih oblik ravninskega kubičnega sistema s singularnostjo tipa center ali fokus v izhodišču. Dokažemo, da so vsi pridobljeni sistemi Darbouxjevo integrabilni. Na koncu raziščemo še orbitalno reverzibilnost teh sistemov. V zadnjem poglavju se posvetimo limitnim ciklom. Opišemo enega ključnih pojavov za nastanek limitnih ciklov, Hopfovo bifurkacijo. Predstavimo metodo preiskovanja točk v neskončnosti, Poincaréjevo kompaktifikacijo in tehniko analize okolice neenostavnih singularnih točk, usmerjeno napihovanje. Nato raziščemo možnosti za pojav limitnih ciklov v tridimenzionalnem biokemičnem modelu in opredelimo fazno sliko v prvem kvadrantu dvodimenzionalnega reakcijskega modela"}],"edm:type":"TEXT","dc:type":[{"@xml:lang":"sl","#text":"visokošolska dela"},{"@xml:lang":"en","#text":"theses and dissertations"},{"@rdf:resource":"http://www.wikidata.org/entity/Q1266946"}]},"ore:Aggregation":{"@rdf:about":"http://www.dlib.si/?URN=URN:NBN:SI:DOC-DE2HVRI0","edm:aggregatedCHO":{"@rdf:resource":"URN:NBN:SI:DOC-DE2HVRI0"},"edm:isShownBy":{"@rdf:resource":"http://www.dlib.si/stream/URN:NBN:SI:DOC-DE2HVRI0/e9d98bccd4630cf597c221-35-e9464-a6c-/PDF"},"edm:rights":{"@rdf:resource":"http://rightsstatements.org/vocab/InC/1.0/"},"edm:provider":"Slovenian National E-content Aggregator","edm:intermediateProvider":{"@xml:lang":"en","#text":"National and University Library of Slovenia"},"edm:dataProvider":{"@xml:lang":"sl","#text":"Univerza v Mariboru, Fakulteta za naravoslovje in matematiko"},"edm:object":{"@rdf:resource":"http://www.dlib.si/streamdb/URN:NBN:SI:DOC-DE2HVRI0/maxi/edm"},"edm:isShownAt":{"@rdf:resource":"http://www.dlib.si/details/URN:NBN:SI:DOC-DE2HVRI0"}}}}