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1.
Racionalne involucije : na študijskem programu 2. stopnje Matematika
Ivan Mastev, 2023, master's thesis

Abstract: V magistrskem delu določamo dvodimenzionalne racionalne involucije s pomočjo določenih metod v računski komutativni algebri. Najprej poiščemo vse dvodimenzionalne racionalne involucije, ki imajo v števcih in imenovalcih polinome stopnje ena, nato pa poiščemo vse dvodimenzionalne racionalne involucije, ki imajo v števcih in imenovalcih polinome stopnje dve. Zapisana je teorija, ki se uporablja pri določanju teh involucij, kot tudi vsi algoritmi, ki omogočajo iskanje rešitev.
Keywords: racionalne involucije, ideal, groebnerjeva baza, afina raznoterost, polinomi
Published in DKUM: 30.08.2023; Views: 224; Downloads: 21
.pdf Full text (533,17 KB)

2.
Integrabilnost, linearizabilnost in limitni cikli polinomskih sistemov navadnih diferencialnih enačb : doktorska disertacija
Barbara Arcet, 2023, doctoral dissertation

Abstract: 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.
Keywords: sistemi navadnih diferencialnih enačb, integrabilnost, linearizabilnost, limitni cikli, reverzibilnost, Hamiltonski sistemi
Published in DKUM: 15.03.2023; Views: 463; Downloads: 58
.pdf Full text (2,58 MB)

3.
Computation of focus quantities of three-dimensional polynomial systems
Valery Romanovski, Douglas Shafer, 2014, original scientific article

Abstract: Fix a collection of polynomial vector fields on $R^3$ with a singularity at the origin, for every one of which the linear part at the origin has two pure imaginary and one non-zero eigenvalue. Some such systems admit a local analytic first integral, which then defines a local center manifold of the system.Conditions for existence of a first integral are given by the vanishing certain polynomial or rational functions in the coefficients of the system called focus quantities. In this paper we prove that the focus quantities have a structure analogous to that in the two-dimensional case and use it to formulate an efficient algorithm for computing them.
Keywords: integrability, focus quantities, center conditions
Published in DKUM: 07.08.2017; Views: 1121; Downloads: 122
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4.
On the solvability of second-order impulsive differential equations with antiperiodic boundary value conditions
Yepeng Xing, Valery Romanovski, 2008, original scientific article

Abstract: We prove existence results for second-order impulsive differential equations with antiperiodic boundary value conditions in the presence of classical fixed point theorems. We also obtain the expression of Green's function of related linear operator in the space of piecewise continuous functions.
Keywords: second-order impulsive differential equations, antiperiodic boundary value conditions, Green's function
Published in DKUM: 26.06.2017; Views: 1260; Downloads: 399
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5.
Periodic solutions and asymptotic analysis of ordinary differential equations : editorial
Maoan Han, Pei Yu, Valery Romanovski, Tonghua Zhang, 2014, preface, editorial, afterword

Keywords: mathematics, periodic solutions, asymptotic analysis
Published in DKUM: 13.06.2017; Views: 925; Downloads: 314
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6.
Bifurcations of planar Hamiltonian systems with impulsive perturbation
Zhaoping Hu, Maoan Han, Valery Romanovski, 2013, original scientific article

Abstract: In this paper, by means of the Melnikov functions we consider bifurcations of harmonic or subharmonic solutions from a periodic solution of a planar Hamiltonian system under impulsive perturbation. We give some sufficient conditions under which a harmonic or subharmonic solution exists.
Keywords: matematika, Hamiltonski sistemi, diferencialne enačbe, bifurkacija, mathematics, Hamiltonian systems, differential equations, bifurcation
Published in DKUM: 10.07.2015; Views: 1353; Downloads: 97
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8.
Limit cycle bifurcations from a nilpotent focus or center of planar systems
Maoan Han, Valery Romanovski, 2012, original scientific article

Abstract: We study analytic properties of the Poincaré return map and generalized focal values of analytic planar systems with a nilpotent focus or center. We use the focal values and the map to study the number of limit cycles of this kind of systems and obtain some new results on the lower and upper bounds of the maximal number of limit cycles bifurcating from the nilpotent focus or center. The main results generalize the classical Hopf bifurcation theory and establish the new bifurcation theory for the nilpotent case.
Keywords: mathematic, limit cycles, bifurcation, center problem
Published in DKUM: 10.07.2015; Views: 1115; Downloads: 330
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9.
Stability and periodic oscillations in the Moon-Rand systems
Adam Mahdi, Valery Romanovski, Douglas Shafer, 2013, original scientific article

Abstract: The Moon-Rand systems, developed to model control of flexible space structures, are systems of differential equations on ▫$RR^3$▫ with polynomial or rational right hand sides that have an isolated singularity at the origin at which the linear part has one negative and one pair of purely imaginary eigenvalues for all choices of the parameters. We give a complete stability analysis of the flow restricted to a neighborhood of the origin in any center manifold of the Moon-Rand systems, solve the center problem on the center manifold, and find sharp bounds on the number of limit cycles that can be made to bifurcate from the singularity when it is a focus. We generalize the Moon-Rand systems in a natural way, solve the center problem in several cases, and provide sufficient conditions for the existence of a center, which we conjecture to be necessary.
Keywords: matematika, cikličnost, integrabilnost, stabilnost, mathematics, cyclicity, integrability, stability
Published in DKUM: 10.07.2015; Views: 1095; Downloads: 91
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10.
Foreword from the editors
Viktor Levandovskyy, Dušan Pagon, Marko Petkovšek, Valery Romanovski, 2012, preface, editorial, afterword

Keywords: matematika
Published in DKUM: 10.07.2015; Views: 851; Downloads: 32
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