20.500.12556/DKUM-81001 Integrabilnost in linearizabilnost persistentnih p : -q resonantnih polinomskih sistemov navadnih diferencialnih enačb Integrability and linearizability of persistent p : -q resonant polynomial systems of ordinary differential equations Eden izmed osrednjih problemov teorije navadnih diferencialnih enačb je problem integrabilnosti, ki je pomemben za razumevanje bistva teorije diferencialnih enačb in za uporabo diferencialnih enačb pri študiju dinamičnih procesov v realnem svetu. Osrednji problemi te doktorske disertacije so problem središča in z njim povezana integrabilnost ter problem linearizabilnosti p:-q resonantnih sistemov s kvadratnimi nelinearnostmi. Obravnavana je tudi povezava med $p:-q$ resonantnimi sistemi in njim pripadajočimi persistentnimi sistemi. V uvodu so predstavljeni osnovni pojmi komutativne računske algebre, s poudarkom na lastnostih polinomskih idealov in njihovih raznoterosti. Predstavljene so tudi normalne forme, problem središča in linearizabilnosti sistema ter postopek kompleksifikacije realnih sistemov. Drugo poglavje je namenjeno posplošitvi pojma p:-q resonantnega središča na persistentno p:-q resonantno središče. Izračunamo vse potrebne in zadostne pogoje za nastop persistentnih središč za pet družin p:-q resonantnih sistemov s kvadratnimi nelinearnostmi. Raziskana je povezava med integrabilnostjo p:-q resonantnih sistemov in integrabilnostjo pripadajočih persistentnih sistemov. V tretjem poglavju so predstavljeni pogoji za nastop linearizabilnega središča za družino 2:-3 resonantnih sistemov, ki jih dobimo z dekompozicijo raznoterosti ideala, generiranega s količinami linearizabilnosti. Zaradi zahtevnosti izračunov uporabimo pristop, ki temelji na modularni aritmetiki. Za dokazovanje zadostnosti tako pridobljenih pogojev uporabimo več različnih metod, najpogosteje uporabimo metodo, ki temelji na Darbouxjevi teoriji linearizabilnosti. V zadnjem poglavju obravnavamo relativno nov problem (šibko) persistentnega linearizabilnega središča, ki ga posplošimo na problem (šibko) persistentnega linearizabilnega p:-q resonantnega središča. Obravnavana je povezava med linearizacijsko transformacijo p:-q resonantnega sistema in linearizacijsko transformacijo ustreznega persistentnega p:-q resonantnega sistema. Povezava je ponazorjena na 1:-2 in 2:-3 resonantnih kvadratnih sistemih. One of the main problems in the theory of ordinary differential equations is the integrability problem, since it gives an important essential insight to theory of differential equations and the way to use differential equations in studies of dynamical processes in real-world systems. The main problems considered in this are the center problem and its closely related integrability and linearizability problems of p:-q resonant systems with quadratic nonlinearities. We also explain the relation between p:-q resonant system and the corresponding persistent p:-q resonant system in the sense of integrability and linearizability problems. In the first chapter, some basic notations of the commutative computational algebra are presented, mainly the properties of polynomial ideals and their varieties. We introduce the center and linearizability problems and describe the complexification of real systems. The second chapter is devoted to the generalization of the notion of a persistent center to a persistent p:-q resonant center. We find all conditions for existence of the persistent p:-q resonant center for several p:-q resonant systems with quadratic nonlinearities. We also prove the relation between integrability of p:-q resonant system and the corresponding persistent p:-q resonant system. In the third chapter, linearizability of 2:-3 resonant system with quadratic nonlinearities is studied. The conditions for linearizability are obtained by computing the ideal generated by the linearizability quantities and its decomposition into associate primes. In order to complete calculations we use an approach based on modular computations. The sufficiency of the obtained conditions is proven by several methods, mainly by the method of Darboux linearization. In the last chapter we define linearizability of a persistent p:-q resonant centers and unveil the link between linearizability transformation for p:-q resonant systems on one hand and linearizability transformation for persistent p:-q resonant systems on the other hand. Finally, the obtained results are demonstrated on persistent 1:-2 and 2:-3 resonant systems with quadratic nonlinearities. polinomski sistem NDE p:-q resonantni sistem problem integrabilnosti problem linearizabilnosti količine integrabilnosti količine linearizabilnosti persistentno resonantno središče prvi integral polynomial system of ODEs p:-q resonant system integrability problem linearizability problem focus quantities linearizability quantities persistent resonant center first integral true true false M. Žulj] Slovenski jezik Angleški jezik Doktorsko delo/naloga 2021-12-06 09:41:21 2022-06-08 09:46:10 2022-08-10 00:15:20 0000-00-00 00:00:00 2022 0 [Maribor 0 0000-00-00 NiDoloceno NiDoloceno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 517.9(043.3) 110849539 DOK_Zulj_Maja_2022.pdf DOK_Zulj_Maja_2022.pdf 1 30EBBF4C04B0FBE2C3EAD7B7A137FD29 7ac096a800383b6fb54a7f9ca3a0bf64c42da9d6f258729ce518866e2c45c144 47175115-e0d5-11ec-9cd9-00155d000105 20.500.12556/dkum/3060adde-05cb-4a28-9026-ebb7dbb39888 https://dk.um.si/Dokument.php?lang=slv&id=158207 Fakulteta za naravoslovje in matematiko 0 0 0