20.500.12556/DKUM-82982 Integrabilnost, linearizabilnost in limitni cikli polinomskih sistemov navadnih diferencialnih enačb doktorska disertacija Integrability, Linearizability and Limit Cycles of Polynomial Systems of Ordinary Differential Equations Krovna tema pri\v cujo\v ce doktorske disertacije je kvalitativna obravnava nekaterih dru\v zin navadnih diferencialnih ena\v cb (NDE). Osrednja pozornost je namenjena ravninskim in tridimenzionalnim polinomskim sistemom ter preiskovanju pogojev, pri katerih se sistemi pona\v sajo s katero od naslovnih lastnosti: integrabilnostjo, linearizabilnostjo ali prisotnostjo limitnih ciklov. Uvodno poglavje je namenjeno definiciji osnovnih pojmov, ki zadevajo singularne to\v cke in njihove okolice v sistemih NDE. Predstavimo nekaj klju\v cnih metod in algoritmov komutativne ra\v cunske 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 na\v cinov za dokaz zadostnosti teh pogojev. Za preu\v citev linearizabilnosti se dotaknemo teorije normalnih form, predstavimo na\v cin za iskanje pogojev za linearizabilnost sistemov in doka\v zemo izrek, ki povezuje integrabilnost ter linearizabilnost sistemov NDE. Z uporabo omenjene teorije nato preu\v cimo integrabilnost in linearizabilnost kvadrati\v cnega tridimenzionalnega sistema z $(1:-1:-1)$-resonantno singularnostjo v izhodi\v s\v cu. 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 koli\v cin, in sicer iskanje polinomske linearizacije ene od ena\v cb sistema. Pri prou\v cevanju linearizabilnosti se osredoto\v cimo na nekatere Hamiltonske sisteme s homogenimi in nehomogenimi nelinearnostmi stopnje kve\v cjemu sedem. V \v cetrtem delu disertacije se lotimo problema centra in fokusa za nekatere rever-zibilne kubi\v cne sisteme. V tem smislu preiskujemo tri sisteme, ki so z ustrezno transformacijo prevedeni v eno izmed kanoni\v cnih oblik ravninskega kubi\v cnega sistema s singularnostjo tipa center ali fokus v izhodi\v s\v cu. Doka\v zemo, da so vsi pridobljeni sistemi Darbouxjevo integrabilni. Na koncu razi\v s\v cemo \v se orbitalno reverzibilnost teh sistemov. V zadnjem poglavju se posvetimo limitnim ciklom. Opi\v semo enega klju\v cnih pojavov za nastanek limitnih ciklov, Hopfovo bifurkacijo. Predstavimo metodo preiskovanja to\v ck v neskon\v cnosti, Poincar\' ejevo kompaktifikacijo in tehniko analize okolice neenostavnih singularnih to\v ck, usmerjeno napihovanje. Nato razi\v s\v cemo mo\v znosti za pojav limitnih ciklov v tridimenzionalnem biokemi\v cnem modelu in opredelimo fazno sliko v prvem kvadrantu dvodimenzionalnega reakcijskega modela. 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. sistemi navadnih diferencialnih enačb integrabilnost linearizabilnost limitni cikli reverzibilnost Hamiltonski sistemi systems of ordinary differential equations integrability linearizability limit cycles reversibility Hamiltonian systems true true false [B. Arcet] Slovenski jezik Angleški jezik Doktorsko delo/naloga 2022-09-12 16:38:43 2023-03-15 16:53:14 2023-03-16 03:09:18 0000-00-00 00:00:00 2023 0 Maribor 0 Maribor V, 140 str. 0000-00-00 NiDoloceno NiDoloceno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 517.91(043.3) 145318659 DOK_Arcet_Barbara_2023.pdf DOK_Arcet_Barbara_2023.pdf 1 F670706EE8F7E6B05A25B0B4073CFA03 370b9a4ced868ae32b739625acb76c8b1c9670fe455597794737fc7d64a99019 66d57067-b74d-11ed-9c0c-00155d000105 https://dk.um.si/Dokument.php?lang=slv&id=165868 Fakulteta za naravoslovje in matematiko 0 0 0