Qualitative studies of some polynomial systems of ordinary differential equations

doctoral dissertation

Maša Dukarić (Author), Valery Romanovski (Mentor), Jaume Giné (Co-mentor)

Abstract

This doctoral dissertation is devoted to the studies of some qualitative properties of certain polynomial systems of ordinary differential equations. The main problems that are considered in this thesis are the problems of integrability and cyclicity. Some results on the classification of the global phase portraits of a family of cubic systems are presented as well. In the first chapter basic notions and results of the qualitative theory of systems of ODE's are introduced. Since one of important tools for our study of these problems is the commutative computational algebra, some main notions and properties of polynomial ideals and their varieties, including various algorithms related to them, are also presented in the introduction. In the second chapter methods for investigation of trajectories near degenerated singularities are presented. They are further used for the classification of global phase portraits of a family of cubic systems with the nilpotent center at the origin. In the third chapter the main problem of these thesis is studied, the problem of integrability. The problem of integrability which is connected to the problem of distinguishing between a center and a focus is studied for two different families of cubic polynomial systems of ODE's. With the computational algebra approach the necessary conditions for the existence of the first integral of these systems were obtained. For all but one condition was proven, using various approaches, the existence of the first integrals. The center problem for the real systems can be generalized to the complex systems. The origin of the system obtained after the complexification of the real system is the so-called 1:-1 resonant singular point, from which one additional generalization follows. This is the generalization to the p:-q resonant center. In the third chapter the :-3 resonant singular point of a quadratic family of complex systems is studied. The fourth chapter is devoted to the study of the problem of integrability of a three dimensional polynomial system with quadratic nonlinearities. The problem of existence of two independent first integrals and the existence of one first integral in the system was investigated. In the last chapter local bifurcations of limit cycles of a family of cubic systems are studied. Estimations for the number of limit cycles bifurcated from each components of the center variety are obtained.

Keywords

planar systems of ODEʼs;higher dimensional system of ODEʼs;phase portrait;nilpotent center;limit cylces;Poincaré compactification;center problem;Bautin ideal;focus quantities;time-reversibility;integrability problem;darboux method;linearizability;limit cycle;cyclicity;dissertations;

Data

Language:	English
Year of publishing:	2016
Typology:	2.08 - Doctoral Dissertation
Organization:	UM FNM - Faculty of Natural Sciences and Mathematics
Publisher:	M. Dukarić]
UDC:	517.9(043.3)
COBISS:	22346760
Views:	1393
Downloads:	195
Average score:	0 (0 votes)
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Other data

Secondary language:	Slovenian
Secondary title:	Kvalitativne študije nekaterih polinomskih sistemov navadnih diferencialnih enačb
Secondary abstract:	Ta doktorska disertacija je posvečena študijam nekaterih kvalitativnih lastnosti določenih sistemov navadnih diferencialnih enačb. Osrednja obravnavana problema te teze sta problem integrabilnosti in cikličnosti. Prav tako so predstavljeni nekateri rezultati o klasifikaciji globalnih faznih portretov družine kubičnih sistemov. V prvem poglavju bodo predstavljeni osnovni pojmi in rezultati kvalitativne teorije sistemov NDE. Ker je pomembno orodje za študijo teh problemov komutativna računska algebra, bodo v uvodu predstavljeni tudi osnovni pojmi in lastnosti polinomskih idealov ter njihovih raznoterosti, vključno s številnimi algoritmi povezanimi s tem. V drugem poglavju je predstavljena metoda za raziskovanje obnašanja trajektorij v okolici izrojenih singularnosti. Ta je uporabljena v nadaljevanju pri klasifikaciji globalnih faznih portretov družine kubičnih sistemov z nilpotentnim centrom v izhodišču. V tretjem poglavju je študiran osrednji problem te disertacije, problem integrabilnosti. Problem integrabilnosti, ki je tesno povezan s problemom razlikovanja med centrom in fokusom, je bil raziskan za dve različni družini kubičnih sistemov NDE. S pristopom, ki temelji na računski algebri, so bili pridobljeni potrebni pogoji obstoja prvega integrala teh sistemov. Za vse razen za en pogoj je bil, s številnimi pristopi, dokazan obstoj prvega integrala. Problem centra realnih sistemov je lahko posplošen na kompleksne sisteme. Izhodišče sistema pridobljenega po kompleksifikaciji realnega sistema je tako imenovana 1:-1 resonantna singularnost, in iz tega sledi še dodatna posplošitev. To je tako imenovana posplošitev na p:-q resonantne centre. V tretjem poglavju je predstavljena tudi študija 1:-3 resonantne singularnosti kvadratične družine kompleksnega sistema. Četrto poglavje je namenjeno študiji problema integrabilnosti tridimenzionalnih sistemov s kvadratičnimi nelinearnostmi. Raziskan je bil problem obstoja dveh neodvisnih prvih integralov ter obstoja enega prvega integrala te družine. V zadnjem poglavju so raziskane bifurkacije limitnih ciklov družine kubičnih sistemov. Pridobljene so ocene števila limitnih ciklov, ki bifurcirajo iz vsake komponente raznoterosti centra.
Secondary keywords:	ravninski sitem NDE;višje dimenzionalni sistemi NDE;fazni portreti;nilpotentni center;limitni cikli;Poincaréjeva kompaktifikacija;problem centra;Bautinov ideal;fokusne količine;problem integrabilnosti;Darbouxjeva metoda;linearizabilnost;limitni cikel;cikličnost;disertacije;
URN:	URN:SI:UM:
Type (COBISS):	Doctoral dissertation
Thesis comment:	Univ. v Mariboru, Fak. za naravoslovje in matematiko, Oddelek za matematiko in računalništvo
Pages:	IX, 149 str.
ID:	9124498