Integrability and local bifurcations in polynomial systems of ordinary differential equations

doctoral dissertation

Brigita Ferčec (Avtor), Valery Romanovski (Mentor), Douglas Shafer (Komentor)

Povzetek

V tej doktorski disertaciji obravnavamo naslednje probleme kvalitativne teorije navadnih diferencialnih enačb (NDE): problem centra in fokusa, problem cikličnosti, problem izohronosti in problem bifurkacij kritičnih period. V prvem poglavju vpeljemo nekaj glavnih pojmov kvalitativne teorije NDE in opišemo nekaj temeljnih metod in algoritmov komutativne računske algebre, ki so potrebni za našo študijo. V drugem poglavju obravnavamo problem razlikovanja med centrom in fokusom, ki je ekvivalenten problemu obstoja prvega integrala določene oblike za dan sistem. To je vzrok, zakaj problemu centra in fokusa pravimo tudi problem integrabilnosti. Poiskali smo potrebne pogoje za integrabilnost (pogoje za center) za družino dvodimenzionalnih kubičnih sistemov, za Lotka-Volterrov sistem v obliki linearnega centra, motenega s homogenimi polinomi četrte stopnje in za nekatere polinomske družine v obliki linearnega centra, motenega s homogenimi polinomi pete stopnje. Z uporabo različnih metod smo za večino teh pogojev dokazali njihovo zadostnost za integrabilnost. Nadalje smo v tretjem poglavju z uporabo metod računske algebre pridobili zgornjo mejo za cikličnost (t.j. število limitnih ciklov, ki bifurcirajo iz izhodišča pri majhnih motnjah) družine kubičnih sistemov, obravnavane v drugem poglavju. Izračune premaknemo v polinomsko podalgebro, ki je povezana s časovno rezerzibilnimi sistemi družine in se na tak način izognemo problemu neradikalnosti Bautinovega ideala, povezanega s tem sistemov. Prav tako določimo število limitnih ciklov, ki bifurcirajo iz vsake komponente raznoterosti centra. V zadnjem poglavju disertacije obravnavamo problem izohronosti in problem bifurkacij kritičnih period za tridimenzionalne sisteme s centralnimi mnogoterostmi, na katerih vse trajektorije ustrezajo periodičnim rešitvam sistema. Za koeficiente sistema podamo kriterije za koeficiente sistema za razlikovanje med primeri izohronih in primeri neizohronih nihanj in za določitev zgornje meje števila kritičnih period.

Ključne besede

system of ODEʼs;integrability;center problem;time-reversibility;Darboux integral;linearizability;center variety;focus quantity;limit cycle;cyclicity problem;bifurkations of critical periods;period function;isochronicity;mathematics;dissertations;

Podatki

Jezik:	Angleški jezik
Leto izida:	2013
Izvor:	[Maribor
Tipologija:	2.08 - Doktorska disertacija
Organizacija:	UM FNM - Fakulteta za naravoslovje in matematiko
Založnik:	B. Ferčec]
UDK:	517.91/.93(043.3)
COBISS:	19968520
Št. ogledov:	1881
Št. prenosov:	179
Ocena:	0 (0 glasov)
Metapodatki:

Ostali podatki

Sekundarni jezik:	Slovenski jezik
Sekundarni naslov:	Integrability and local bifurcations in polynomial systems of ordinary differential equations
Sekundarni povzetek:	In this doctoral dissertation we study the following problems of the qualitative theory of ordinary differential equations (ODE's): the center-focus problem, the cyclicity problem, the isochronicity problem and the problem of bifurcations of critical periods. In the first chapter we introduce a few main notions of the qualitative theory of ODE's and describe some basic methods and algorithms of commutative computational algebra, which are needed for our study. In the second chapter we consider the problem of distinguishing between a center and a focus, which is equivalent to the problem of existence for the system of a first integral of a certain type. This is why the center-focus problem is also called the integrability problem. We find necessary conditions of integrability (the center conditions) for a family of two-dimensional cubic systems, for a Lotka-Volterra system in the form of the linear center perturbed by homogeneous quartic polynomials, and for some polynomial families in the form of a linear center perturbed by fifth degree homogeneous polynomials. Using various methods we prove that most of the necessary conditions obtained are also sufficient conditions for integrability. Then, in the third chapter using methods of computational algebra we obtain an upper bound for the cyclicity (that is, the number of limit cycles bifurcating from the origin after small perturbations) of the family of cubic systems studied in the previous chapter. We overcome the problem of nonradicality of the associated Bautin ideal by moving computations to a polynomial subalgebra associated with time-reversible systems of the family. We also determine the number of limit cycles bifurcating from each component of the center variety. In the last chapter of the thesis we consider the problems of isochronicity and bifurcations of critical periods for three-dimensional systems with center manifolds filled with closed trajectories. We give criteria on the coefficients of the system to distinguish between the cases of isochronous and non-isochronous oscillations and to determine an upper bound on the number of critical periods.
Sekundarne ključne besede:	sistem NDE;integrabilnost;problem centra;časovna reverzibilnost;Darbouxov integral;linearizabilnost;raznoterost centra;fokusna količina;limitni cikel;bifurkacije kritičnih period;funkcija periode;izohronost;matematika;disertacije;
URN:	URN:SI:UM:
Vrsta dela (COBISS):	Doktorsko delo/naloga
Komentar na gradivo:	Univ. v Mariboru, Fak. za naravoslovje in matematiko, Oddelek za matematiko in računalništvo
Strani:	X, 117 str.
Ključne besede (UDK):	mathematics;natural sciences;naravoslovne vede;matematika;mathematics;matematika;analysis;matematična analiza;differential equations;integral equations;other functional equations;finite differences;calculus of variations;functional analysis;diferencialne enačbe;integralske enačbe;druge funkcionalne enačbe;končne diference;variacijski račun;funkcionalna analiza;
ID:	81847