INTEGRABILITY AND LOCAL BIFURCATIONS IN POLYNOMIAL … · cikliˇcnosti, bifurkacije kritiˇcnih...

UNIVERSITY OF MARIBORFACULTY OF NATURAL SCIENCES AND MATHEMATICS

DOCTORAL DISSERTATION

INTEGRABILITY AND LOCAL BIFURCATIONS

IN POLYNOMIAL SYSTEMS OF ORDINARY

DIFFERENTIAL EQUATIONS

June, 2013 Brigita Fercec

UNIVERSITY OF MARIBORFACULTY OF NATURAL SCIENCES AND MATHEMATICS

DOCTORAL DISSERTATION

INTEGRABILITY AND LOCAL BIFURCATIONS

IN POLYNOMIAL SYSTEMS OF ORDINARY

DIFFERENTIAL EQUATIONS

June, 2013 Brigita FercecMentor: Dr. Valerij Romanovskij

Co-mentor: Prof. Dr. Douglas S. Shafer

UNIVERZA V MARIBORUFAKULTETA ZA NARAVOSLOVJE IN MATEMATIKO

DOKTORSKA DISERTACIJA

INTEGRABILNOST IN LOKALNE BIFURKACIJE

V POLINOMSKIH SISTEMIH NAVADNIH

DIFERENCIALNIH ENACB

Junij, 2013 Brigita FercecMentor: Dr. Valerij Romanovskij

Somentor: Prof. Dr. Douglas S. Shafer

To my family, especially my mother who has given me a uniqueexample of persistence and patience.

Moji druzini, se posebej mami, ki mi daje enkratenzgled vztrajnosti in potrpezljivosti.

Contents

Abstract iv

Povzetek vi

Acknowledgement viii

1 Introduction 1

1.1 Singular points and limit cycles of planar systems of ODE’s . . . . . . . . . . . . 1

1.1.1 Autonomous systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.2 Linear systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.3 Nonlinear systems in the plane . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 Polynomial ideals and their varieties . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2.1 Grobner basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.2.2 Operations on ideals and varieties . . . . . . . . . . . . . . . . . . . . . . 15

1.2.3 Decomposition of varieties using modular arithmetics . . . . . . . . . . . 17

2 The problems of center and local integrability 21

2.1 The Poincare first return map and the center problem . . . . . . . . . . . . . . . 22

2.2 Complexification of the system, the center variety and focus quantities . . . . . . 25

2.3 Time-reversible and Hamiltonian systems, Darboux method . . . . . . . . . . . . 29

2.3.1 Time-reversibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.3.2 Hamiltonian systems and Darboux integrability . . . . . . . . . . . . . . . 32

2.4 Local integrability of a subfamily of the cubic system . . . . . . . . . . . . . . . . 34

2.5 Local integrability of a quartic Lotka–Volterra system . . . . . . . . . . . . . . . 40

2.6 Local integrability of a quintic system . . . . . . . . . . . . . . . . . . . . . . . . 48

3 Bifurcations of limit cycles 55

3.1 The cyclicity problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.2 The cyclicity of the subfamily of cubic systems . . . . . . . . . . . . . . . . . . . 61

3.2.1 Cyclicity of the components . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4 Bifurcations of critical periods 67

4.1 The period function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.2 An approach to studying isochronicity and bifurcations of critical periods . . . . 70

4.2.1 The isochronicity problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.2.2 The linearizability problem . . . . . . . . . . . . . . . . . . . . . . . . . . 72

iii

4.3 Bifurcations of critical periods for some families of polynomial systems . . . . . . 754.3.1 Center manifols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.3.2 Isochronicity and critical periods of a three-dimensional quadratic system 78

Appendix A 87

Appendix B 89

Appendix C 91

Appendix D 93

Bibliography 97

List of Figures 103

Razsirjeni povzetek 105

Professional Curriculum Vitae 115

iv

Abstract

In this doctoral dissertation we study the following problems of the qualitative theory of ordinarydifferential equations (ODE’s): the center-focus problem, the cyclicity problem, the isochronicityproblem and the problem of bifurcations of critical periods. In the first chapter we introducea few main notions of the qualitative theory of ODE’s and describe some basic methods andalgorithms of commutative computational algebra, which are needed for our study. In thesecond chapter we consider the problem of distinguishing between a center and a focus, whichis 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 necessaryconditions 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 quarticpolynomials, and for some polynomial families in the form of a linear center perturbed by fifthdegree homogeneous polynomials. Using various methods we prove that most of the necessaryconditions obtained are also sufficient conditions for integrability. Then, in the third chapterusing 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 familyof cubic systems studied in the previous chapter. We overcome the problem of nonradicality ofthe associated Bautin ideal by moving computations to a polynomial subalgebra associated withtime-reversible systems of the family. We also determine the number of limit cycles bifurcatingfrom each component of the center variety. In the last chapter of the thesis we consider theproblems of isochronicity and bifurcations of critical periods for three-dimensional systems withcenter manifolds filled with closed trajectories. We give criteria on the coefficients of the systemto distinguish between the cases of isochronous and non-isochronous oscillations and to determinean upper bound on the number of critical periods.

Math. Subj. Class. (2010): 34C05, 34C07, 34C23, 34C25, 34C45, 37G15.UDK: 517.91/.93(043.3)

Keywords: system of ODE’s, integrability, center problem, time-reversibility, Darboux inte-gral, linearizability, center variety, focus quantity, limit cycle, cyclicity problem, bifurcations ofcritical periods, period function, isochronicity problem.

v

Povzetek

V tej doktorski disertaciji obravnavamo naslednje probleme kvalitativne teorije navadnih difer-encialnih enacb (NDE): problem centra in fokusa, problem ciklicnosti, problem izohronosti inproblem bifurkacij kriticnih period. V prvem poglavju vpeljemo nekaj glavnih pojmov kval-itativne teorije NDE in opisemo nekaj temeljnih metod in algoritmov komutativne racunskealgebre, ki so potrebni za naso studijo. V drugem poglavju obravnavamo problem razlikovanjamed centrom in fokusom, ki je ekvivalenten problemu obstoja prvega integrala dolocene oblike zadan sistem. To je vzrok, zakaj problemu centra in fokusa pravimo tudi problem integrabilnosti.Poiskali smo potrebne pogoje za integrabilnost (pogoje za center) za druzino dvodimenzionalnihkubicnih sistemov, za Lotka-Volterrov sistem v obliki linearnega centra, motenega s homogenimipolinomi cetrte stopnje in za nekatere polinomske druzine v obliki linearnega centra, motenegas homogenimi polinomi pete stopnje. Z uporabo razlicnih metod smo za vecino teh pogojevdokazali njihovo zadostnost za integrabilnost. Nadalje smo v tretjem poglavju z uporabo metodracunske algebre pridobili zgornjo mejo za ciklicnost (t.j. stevilo limitnih ciklov, ki bifurcirajoiz izhodisca pri majhnih motnjah) druzine kubicnih sistemov, obravnavane v drugem poglavju.Izracune premaknemo v polinomsko podalgebro, ki je povezana s casovno rezerzibilnimi sistemidruzine in se na tak nacin izognemo problemu neradikalnosti Bautinovega ideala, povezanegas tem sistemov. Prav tako dolocimo stevilo limitnih ciklov, ki bifurcirajo iz vsake komponenteraznoterosti centra. V zadnjem poglavju disertacije obravnavamo problem izohronosti in prob-lem bifurkacij kriticnih period za tridimenzionalne sisteme s centralnimi mnogoterostmi, nakaterih vse trajektorije ustrezajo periodicnim resitvam sistema. Za koeficiente sistema podamokriterije za koeficiente sistema za razlikovanje med primeri izohronih in primeri neizohronihnihanj in za dolocitev zgornje meje stevila kriticnih period.

Math. Subj. Class. (2010): 34C05, 34C07, 34C23, 34C25, 34C45, 37G15.UDK: 517.91/.93(043.3)

Kljucne besede: sistem NDE, integrabilnost, problem centra, casovna reverzibilnost, Dar-bouxov integral, linearizabilnost, raznoterost centra, fokusna kolicina, limitni cikel, problemciklicnosti, bifurkacije kriticnih period, funkcija periode, problem izohronosti.

vii

Acknowledgments

There are many people who have walked with me through the time of my study and haveconstantly encouraged and advised me. They helped me to overcome many difficulties and bydoing so contributed to the completion of this thesis.

It is right to begin by expressing my most special thanks to my mentor Dr. Valerij Ro-manovskij. With his enthusiasm and dedication to science he has helped me to understand eventhe most difficult concepts of the theory I studied. For all the valuable and lengthy discussionshe had with me, for his patience and perseverance, the international experiences that he ar-ranged, and above all the motivation that he has constantly given me, I am sincerely thankful.He gave me a good example of a strong, hardworking, precise, and upright teacher. Perhaps atsome moments I did not realize the purpose of his severity but today I see that he has given memore than I expected.

I thank Prof. Dr. Marko Robnik for pleasant working conditions at the CAMTP (Centerfor Applied Mathematics and Theoretical Physics) and at academic meetings. His concern wasof great importance during my PhD study and my time at the CAMTP.

Special thanks is devoted also to the members of CAMTP (Thanos Manos, Masa Dukaric,Dimitris Andresas, Yonghui Xia) for all the times of laughter and good will. I would especiallylike to thank Benjamin Batistic for all his help and George Papamikos with whom I shared awonderful time participating together at a conference in Romania. I shared with both of themwonderful and fruitful discussions on science and life.

In the summer of 2011 I spent a wonderful time at the University of North Carolina atCharlotte, USA. Maybe that time some things seemed to be obvious but today I am happy thatI had opportunity to work with my co-mentor Prof. Dr. Douglas S. Shafer. His hospitality,calmness, and kindness made me feel at home even though I was far away. I am also gratefulfor his explanations and mathematical discussions and especially for arranging good living con-ditions in Charlotte. I strongly feel the need to thank two special friends whom I met duringthat time, Dr. Adam Mahdi and Prof. Dr. Jaya Bishwal. Our walks through the beautifulcampus, our long discussions and carefree moments were a valuable experience which helpedme to understand many things for the first time. I am thankful to Adam for all his advice,about research and other matters, and for teaching me how to listen to people. I thank Jayashowing me how to believe in doing good things. Special thanks is devoted to the Departmentof Mathematics and Statistics at UNC Charlotte for providing such good working conditions.

I am grateful for the hospitality that I received at Shanghai Normal University, SichuanUniversity in Chengdu, and at Guilin University of Electronic Tehnology. I wish to expressspecial thanks to Prof. Dr. Maoan Han, Prof. Dr. Weinian Zhang, Dr. Xingwu Chen, andProf. Dr. Wentao Huang. I thank also their students and collegues for lovely moments.

For the time in Russia, especially for the nice and unexpected view of beautiful Moscow and

ix

all care during the conference in Nalchik I am very thankful to Prof. Dr. Dusan Pagon. Forjoint moments when we shared scientific and also everyday discussions I owe thanks to Prof.Dr. Matej Mencinger, who also provided good working conditions at the Civil EngineeringFaculty during my pedagogical work. I would also like to thank members of the Departmentof Mathematics and Computer Science in the Faculty of Natural Science and Mathematics inMaribor for a pleasant experience of giving and sharing my mathematical knowledge with youngmathematicians.

There are many friends with whom I have had the opportunity to spend some time the lastfew years. They showed me that a physical distance between us will never be an obstacle toour friendship. For all those funny moments, honest conversations, moments filled with joy andlaughter, but most of all for the moments that got me over a sadness or fear that I never wouldhave conquered alone, I am happy to have in my life such special friends as Tadeja Levicnik,Nina Hebar, Tamara Korosec, Metka Markoc, Melita Munic, and Mario Fijacko. I feel that Iowe them more than just a statement of thanks.

Most importantly, I cannot forget the huge contribution of my family, especially my motherand my father, for their support and loving thoughts. For every advice and encouragement theygave me I would like to thank my sister Metka and my brother Jani. They will always representmy triangle of good confidence and without them some puzzles of my life would be uncompleted.

”Happiness is to love and to be loved.”

I have gratefully experienced the feeling of loving and of being loved. THANK YOU.Finally, I wish to thank the Slovenian Research Agency (ARRS) for their financial support

during my research at CAMTP.

x

Chapter 1

Introduction

In the first section of this chapter we introduce some key ideas and notions of the qualitative the-ory of ODE’s. The main subject of our consideration are two-dimensional autonomous systems.Then, in the second section we discuss the basic concepts of the theory of polynomial ideals anddescribe the main properties of their varieties. We introduce the concept of Grobner basis andcertain fundamental techniques and algorithms of computational algebra for the investigationof polynomial ideals and their varieties, which will be used for our study of systems of ODE’s.

1.1 Singular points and limit cycles of planar systems of ODE’s

Let X(t,x) be a vector function of variables t and x, with the domain of definition D ⊆ R×Rn

(or D ⊆ C× Cn),

X(t,x) = (X1(t, x1, . . . , xn), . . . , Xn(t, x1, . . . , xn))T ,

x = (x1, . . . , xn)T and x = (dx1dt , . . . ,

dxndt )T . A function x(t), with t in some open interval I ⊆ R,

which satisfies

x(t) =dx

dt= X(t,x(t)) (1.1)

is said to be a solution of the differential equation (1.1).

A necessary condition for x(t) to be a solution is that (t,x(t)) ∈ D for each t ∈ I. If x(t),with domain I, is a solution to (1.1) then so is its restriction to any interval J ⊂ I. If I isthe largest interval for which x(t) satisfies (1.1) then a solution with this property is called themaximal solution.

The existence of solution is determined by the properties of X(t,x). The proofs of followingtwo propositions can be found in [4].

Proposition 1.1.1. If Xi, i = 1, . . . , n are continuous functions in an open domain, D′ ⊆ D ⊆

Rn (or Cn), then given any point (t0,x0) ∈ D′there exists a solution x(t), t ∈ I, of x = X(t,x)

such that t0 ∈ I and x(t0) = x0.

It can happen that system (1.1) with the initial condition x(t0) = x0 has more than onesolution. The following proposition gives a sufficient condition for the uniqueness of the solutionto the initial problem.

1

2 1 Introduction

Proposition 1.1.2. If X and ∂X/∂xi are continuous in an open domain D′ ⊆ D, then given

any (t0,x0) ∈ D′there exists a unique solution x(t) of x = X(t,x) such that x(t0) = x0.

A solution x(t) of x = X(t,x) is presented geometrically by the graph of x(t). This graphdefines a solution curve in the t,x-space. If X is continuous in D, then Proposition 1.1.1 impliesthat the solution curves fill the region D of the t,x-space. This follows because each point in Dmust lie on at least one solution curve. The solutions of the differential equation are, therefore,represented by a family of solution curves in D. If X and ∂X/∂xi are continuous in D thenProposition 1.1.2 implies that there is a unique solution curve passing through every point ofD, that is, the curves filling D do not intersect.

1.1.1 Autonomous systems

In this work we will study the so-called autonomous systems of ODE’s, which are systems ofthe form

x = X(x), x ∈ Rn or x ∈ Cn. (1.2)

The space Rn (Cn) is called the phase space of system (1.2).

System (1.2) is said to be autonomous, because x is determined by x alone, that is, does notdepend on time t, and so the solutions are self-governing. The qualitative behavior of solutioncurves is determined by X(x). When X(c) = 0, the solution x(t) ≡ c is represented by thepoint x = c. These solutions are called fixed points of the system, because x remains at c forall t (such points are also called singular or equilibrium points).

In this work we will mainly deal with two-dimensional autonomous systems, that is, withsystems of the form

x1 = X1(x1, x2), x2 = X2(x1, x2). (1.3)

A solution to (1.3) consists of a pair of functions (x1(t), x2(t)), t ∈ I ⊆ R, which satisfy (1.3).In general, both x1(t) and x2(t) involve arbitrary constants, so that there is a two-parameterfamily of solutions.

The qualitative behavior of this family is determined by how x1(t) and x2(t) behave withchange of t. It is convenient to indicate how x(t) behaves in the phase plane, that is in x1, x2-plane. The qualitative behavior is represented by a family of curves, directed with increasing t.These curves are called trajectories or orbits of system (1.3). The geometrical representation ofthe qualitative picture of trajectories of system (1.3) is called its phase portrait.

1.1.2 Linear systems

A system x = X(x), where x ∈ Rn, is called a linear system of dimension n, if X : Rn → Rn isa linear mapping, that is,

X(x) =

X1(x1, . . . , xn)...

Xn(x1, . . . , xn)

=

a11 · · · a1n...

...an1 · · · ann

x1

...xn

.Thus, the system can be written as

x = Ax

1.1 Singular points and limit cycles of planar systems of ODE’s 3

where A is the coefficients matrix. It is well known (see [4]) that only a finite number ofqualitatively different phase portraits can arise for n-dimensional linear systems. By a linearchange of coordinates x=My each linear system x = Ax can be transformed to the systemy = By, where the matrix B=M−1AM is similar to A and M is a non-singular matrix whosecolumns are linearly independent. Similarity is an equivalence relation on the set of n×nmatricesand it is known that this set can be disjointly decomposed into equivalence or similarity classes.For any two matrices A and B in the same similarity class, the solutions of the systems x = Axand y = By are related by x = My if B=M−1AM. For each positive integer n, there areinfinitely many similarity classes of n×n real matrices. These similarity classes can be groupedinto just finitely many types. We will discuss only the case of real two-dimensional systems,that is, systems of the form

x1 =a11x1 + a12x2

x2 =a21x1 + a22x2.(1.4)

As is well known if A =

[a11 a12a21 a22

]is a real matrix, then there is a non-singular matrix (with

entries in C in case (d) below) M such that the Jordan form J=M−1AM of A is one of thetypes:

(a)

[λ1 00 λ2

], λ1 > λ2;

(b)

[λ0 00 λ0

];

(c)

[λ0 10 λ0

];

(d)

[α+ iβ 0

0 α− iβ

], β > 0.

In case (d) there exists also a matrix M such that J = M−1AM has the form[α −ββ α

].

We see that in the two-dimensional case the matrix A has either two different real eigenvalues λ1and λ2, or one double real eigenvalue λ0, or complex eigenvalues α± iβ. Thus, the elementaryfixed point at the origin of system (1.4) is one of the following four types:

• node - the same signed real eigenvalues

(a) stable if λ2 < λ1 < 0 or λ0 < 0

(b) unstable if λ1 > λ2 > 0 or λ0 > 0

• saddle - different signed real eigenvalues

• focus - complex eigenvalues α± iβ

(a) stable if α < 0

4 1 Introduction

(b) unstable if α > 0

• center - pure imaginary eigenvalues ±iβ.

Qualitative (topological) behavior of trajectories for these singular points (phase portraits) isshown in Figure 1.1.

Figure 1.1: Phase portraits of singular points of linear systems: (a) saddle, (b) unstable node,(c) unstable focus, (d) center (up to linear transformation).

1.1.3 Nonlinear systems in the plane

We now consider autonomous systems of the form

x = X(x), x ∈ R2, (1.5)

where X is a continuously differentiable, nonlinear vector function. Unlike the case of linearsystems these phase portraits are very seldom determined by the nature of the fixed points ofthe system.

We first discuss the behavior of trajectories of nonlinear system in a neighborhood of asingular point. By a linear transformation any point can be moved to the origin of the coordinatesystem, so without lose of generality we consider a system with a singular point at the originwhich is written in the form

x1 = ax1 + bx2 + g1(x1, x2),

x2 = cx1 + dx2 + g2(x1, x2),(1.6)

where [gi(x1, x2)/r] → 0 as r =√x21 + x22 → 0.

The linear system

x1 = ax1 + bx2, x2 = cx1 + dx2 (1.7)

is called the linearization of system (1.6) at the origin. The components of the linear vectorfield (1.7) form the linear part of (1.6).

A fixed point of system (1.6) is called an elementary point if the determinant of the matrix ofthe linearized system is not equal to zero. The next theorem (see e.g. [4, 27]) relates the phaseportrait of a nonlinear system in a neighborhood of a fixed point to that of its linearization.


Theorem 1.1.3 (Linearization theorem). If x = 0 is an elementary fixed point of system (1.6)then in a neighborhood of the origin the phase portraits of (1.6) and its linearization have a fixedpoint of the same type at the origin provided the linearized system is not a center.

The essence of the Linearization theorem is that the center is the case in which the qualitativeequivalence of the nonlinear system and its linearization cannot be read from the linearization.Thus, if the eigenvalues of the linearized system have nonzero real parts, then the phase portraitsof the nonlinear system and its linearization are qualitatively (topologically) equivalent in aneighborhood of the fixed point.

Any point in the phase plane of system (1.5) which is not a fixed point is called an ordinarypoint. If x0 is an ordinary point then X(x0) = 0, and since X is continuous, there is a neigh-borhood of x0 containing only ordinary points. This means that the local phase portrait at theordinary point has no fixed points. The next theorem establishes the qualitative equivalence ofsuch local phase portraits.

Theorem 1.1.4 (Flow box theorem). In a sufficiently small neighborhood of an ordinary pointx0 of system (1.5) there is a differentiable change of coordinates y = y(x) which transforms(1.5) to the system y1 = 0, y2 = 1.

Proofs of the Flow box theorem can be found in [4, 27].The Flow box theorem assures that the local phase portraits at ordinary points are all

qualitatively equivalent since it guarantees the existence of new coordinates with the propertydescribed in theorem, at least in some neighborhood of any ordinary point of system (1.5).

In addition to singular points another important element for characterization of phase por-traits of autonomous systems of ODE’s is a limit cycle.

Definition 1.1.5. A limit cycle of system (1.5) is an isolated closed (periodic) orbit whichcorresponds to a periodic solution of (1.5).

Limit cycles of different types are presented in Figure 1.2.

Figure 1.2: (a) stable limit cycle, (b) unstable limit cycle, (c)–(d) semistable limit cycles.

The α- (respectively ω-) limit set Lα(x) (resp., Lω(x)) is the set of all point p for whichthere exists monotonic sequence tk tending to −∞ (resp., +∞) such that limk→∞ x(tk) = p.

6 1 Introduction

In general, we encounter limit sets that are fixed points or closed orbits. The next theorem(whose proof can be found in [4]) will give the answer to the question to what other possibilitiescan occur. It holds only for planar phase portraits.

Theorem 1.1.6 (Poincare - Bendixson). A non-empty, compact limit set of a phase flow in theplane that does not contain a fixed point is a closed orbit.

It follows from this theorem that if a closed orbit C is a subset of Lα(x) or Lω(x) for somex that does not lie in C, then C is a limit cycle.

Besides fixed points and periodic orbits, a dynamical system can have homoclinic orbits(loops) and heteroclinic orbits. They furnish examples of separatrix cycles and compound sep-aratrix cycles for planar dynamical systems. Separatrix cycles are closed orbits resulting fromsaddle - saddle connection. We describe these two types of orbits using the following examples(see [47]). A system

x = y

y = x+ x2

with a first integral H(x, y) = y2/2 − x2/2 − x3/3 has trajectories lying in curves defined by

y2 − x2 − 2

3x3 = C.

The phase portrait for this system is shown in Figure 1.3. The curve y2 − x2 − 2x3/3 = 0goes through the point (−3/2, 0) and has the saddle at the origin (which also corresponds toC = 0) as its α and ω-limit sets. The solution curve Γ is called a homoclinic orbit and the flowon the simple closed curve determined by the union of this homoclinic orbit and the fixed pointat the origin is called a separatrix cycle.

Figure 1.3: A homoclinic orbit Γ, which defines a separatrix cycle.

Let us construct now the phase portrait for the undamped pendulum for which differentialequations are written as the system

x = y

y = − sinx.(1.8)


The phase portrait of system (1.8) is shown in Figure 1.4. The trajectory Γ1 having thesaddle (−π, 0) as its α-limit set and the saddle at (π, 0) as its ω-limit set is called a heteroclinicorbit. The trajectory Γ2 is also a heteroclinic orbit. The flow on the simple closed curveS = Γ1 ∪Γ2 ∪(π, 0)∪ (−π, 0) defines a separatrix cycle. Each of the curves Γ1 and Γ2 givesan example of a separatrix connection.

Figure 1.4: Heteroclinic orbits Γ1 and Γ2 defining a separatrix cycle.

A finite union of compatibly oriented separatrix cycles is called a compound separatrix cycle.Some examples of compound separatrix cycles are given in Figure 1.5.

It is known that for a planar polynomial system any ω-limit set is either a fixed point, a limitcycle or a union of separatrix cycles. To determine the phase portrait of system (1.5) one needsto establish the types of all singular points of the system, find limit cycles, separatrix cycles andthe trajectories connecting singular points.

8 1 Introduction

Figure 1.5: Examples of compound separatrix cycles.

1.2 Polynomial ideals and their varieties

In this section following mainly [23] and [56] we recall some properties of polynomial ideals andtheir varieties and describe a few main algorithms of computational commutative algebra, whichwe will use for our study of systems of ODE’s in the following chapters.

A polynomial in variables x1, x2, . . . , xn with coefficients in a field k is a formal expressionof the form

f =∑α∈S

aαxα, (1.9)

where S is a finite subset of Nn0 , aα ∈ k, and for α = (α1, α2, . . . , αn), x denotes the monomial

xα11 , xα2

2 · · ·xαnn . The product aαx

α is called a term of the polynomial f . The set of all polyno-mials in the variables x1, x2, . . . , xn with coefficients in k is denoted by k[x1, x2, . . . , xn]. Withthe natural operations of addition and multiplication, k[x1, x2, . . . , xn] is a commutative ring.The full degree of a monomial xα is the number |α|= α1 + α2 + · · · + αn. The full degree of apolynomial f as in (1.9), denoted by deg(f), is the maximum of |α| among all monomials (withnonzero coefficients aα) of f .

We will denote by k a field and by n a natural number. The space

kn = (a1, a2, . . . , an) : a1, a2, . . . , an ∈ k

is called n-dimensional affine space.

Definition 1.2.1. Let f1, . . . , fs be polynomials of k[x1, x2, . . . , xn]. The affine variety definedby the polynomials f1, . . . , fs is the set

V(f1, . . . , fs) = (a1, a2, · · · , an) ∈ kn : fj(a1, a2, · · · , an) = 0 for 1 ≤ j ≤ s.

A subvariety of V is a subset of V that is itself an affine variety.

1.2 Polynomial ideals and their varieties 9

It is clear that there are many collections of polynomials defining the same variety. Tounderstand the concept of variety better we need the notion of an ideal.

Definition 1.2.2. An ideal in the polynomial ring k[x1, x2, . . . , xn] is a subset I ofk[x1, x2, . . . , xn] satisfying

(i) 0 ∈ I,

(ii) if f, g ∈ I then f + g ∈ I, and

(iii) if f ∈ I and h ∈ k[x1, x2, . . . , xn], then hf ∈ I.

Let f1, f2, . . . , fs be polynomials of k[x1, x2, . . . , xn]. We denote

⟨f1, f2, . . . , fs⟩ = s∑

j=1

hjfj : h1, . . . hs ∈ k[x1, x2, . . . , xn]. (1.10)

It is easily seen that ⟨f1, f2, . . . , fs⟩ is an ideal in k[x1, x2, . . . , xn]. Polynomials f1, . . . , fs, arecalled generators of this ideal. Ideal I ⊂ k[x1, x2, . . . , xn] is finitely generated if there ex-ist polynomials f1, f2, . . . , fm ∈ k[x1, x2, . . . , xn] such that I = ⟨f1, f2, . . . , fm⟩ and the setf1, f2, . . . , fs is called a basis of I. For the proof of next theorem one can see [23].

Theorem 1.2.3 (Hilbert Basis Theorem). If k is a field, then every ideal in the polynomial ringk[x1, x2, . . . , xn] is finitely generated.

A direct consequence of Theorem 1.2.3 is that every ascending chain of ideals I1 ⊂ I2 ⊂ I3 ⊂· · · in a polynomial ring over a field k stabilizes. That is, there exists m ≥ 1 such that for everyj > m, Ij = Im. Rings in which every strictly ascending chain of ideals stabilizes are calledNoetherian rings. It is easily seen that if f1, . . . , fs and g1, . . . , gm are bases of an ideal I ⊂k[x1, x2, . . . , xn], that is, I = ⟨f1, . . . , fs⟩ = ⟨g1, . . . , gm⟩, then V(f1, . . . , fs) = V(g1, . . . , gm).

As we have seen any finite or infinite collection of polynomials defines a variety. Conversely,given a variety V , there is a naturally associated to it an ideal called the ideal of the variety Vdefined by

I(V ) = f ∈ k[x1, x2, . . . , xn] : f(a1, . . . , an) = 0 for all (a1, . . . , an) ∈ V .

From the definition of affine variety we see, that to find an affine variety V(f1, . . . , fs) ⊂ kn

is to find the set of solutions of the system

f1(x1, . . . , xn) = 0,

............................

fs(x1, . . . , xn) = 0.

(1.11)

This is a problem that frequently arises in studies of various phenomena in physical, technicaland other sciences. In particular as it was mentioned in the previous section, in order to studythe qualitative picture of trajectories of the autonomous differential system

x1 = f1(x1, . . . , xn),

.................................

xn = fn(x1, . . . , xn)

(1.12)

10 1 Introduction

we first have to determine singular points of the system, which are the points where all polyno-mials f1, . . . fn vanish.

The problem of finding solutions of system (1.11) is very difficult. It can happen that system(1.11) has infinitely many solutions, and then it is simply impossible to find all these solutionsnumerically. Even if system (1.11) has a finite number of solutions it is still very difficult andoften impossible to find all of them numerically without applying methods of computationalalgebra, which we describe below.

1.2.1 Grobner basis

It appears no regular methods to solve a generic system of the form (1.11) were known until themid-sixties of the last century, when Bruno Buchberger (see [8]) invented the theory of Grobnerbases, which is now the cornerstone of modern computational algebra. In this subsection wefirst describe briefly the notion of Grobner basis, which will be used to analyse the structure ofpolynomial ideas and their varieties. Since a Grobner basis of I depends on a term ordering onmonomials of k[x1, . . . , xn] we define the three most important term orders.

Let α = (α1, . . . , αn) and β = (β1, . . . , βn) be elements of Nn0 (N0 = N ∪ 0).

• Lexicographic order. Define α >lex β if and only if, reading left to right, the first nonzeroentry in the n-tuple α− β ∈ Zn is positive.

• Degree lexicographic order. Define α >deglex β if and only if

|α|=n∑

j=1

αj > |β|=n∑

j=1

βj or |α|= |β| and α >lex β .

• Degree-reverse lexicographic order. Define α >drevlex β if and only if

|α|=n∑

j=1

αj > |β|=n∑

j=1

βj

or|α|= |β| and the rightmost nonzero entry of α− β ∈ Zn is negative.

For example, if α = (3, 2, 5, 2) and β = (1, 6, 0, 5), then α is greater than β with respect toall three orders. An ordering on N0 induces an ordering on monomials of k[x1, . . . , xn]. Thedefinitions above are based on the presumed ordering x1 > · · · > xn of the variables. If in k[x, y]we choose x > y, then x3y >lex xy

4 (since (3, 1) >lex (1, 4)), x3y <deglex xy4 (since 3 + 1 < 1 + 4

yielding (3, 1) <deglex (1, 4)) and x3y >drevlex xy4 (since (3, 1) >drevlex (1, 4)).

Now we describe the procedure of division of a polynomial by an ordered set of polynomials,that is, to divide f ∈ k[x1, . . . , xn] by an ordered set F = f1, . . . , fs, which means to expressf in the form

f = a1f1 + · · · asfs + r, (1.13)

where the quotients a1, . . . , an and the remainder r are polynomials of k[x1, . . . , xn], and eitherr = 0 or deg(r) < deg(f). If (1.13) and the properties mentioned above hold, we say that f isreduced to r modulo F and write

fF−→ r.


The process depends on the monomial order and the order of the polynomials in the set F =f1, . . . , fs. So, we have to fix this order to perform the division. We will not give precisedefinitions (which can be found in [1, 23, 56]) but we demonstrate the division process by thefollowing example.

Example 1.2.4. Let us divide f = xy2+1 by f1 = xy−1 and f2 = y2−1 using the lexicographicorder with x > y.

u1 : x+ yu2 : 1 r

f1 : xy − 1√x2y + xy2 + y2

f2 : y2 − 1x2y − x

xy2 + x+ y2

xy2 − yx+ y2 + y

y2 + y → xy2 − 1

y + 11 → x+ y0 → x+ y + 1

On the first step the leading term LT (f1) = xy divides the leading term LT (f) = x2y. Thuswe divide x2y by xy, leaving x and then subtract x · f1 from f . Next, we repeat the sameprocess on xy2 + x + y2. We divide again f by LT (f1) = xy. Note that neither LT (f1) = xynor LT (f2) = y2 divides LT (x + y2 + y) = x. However, x + y2 + y is not the remainder sinceLT (f2) divides y2. Thus, if we move x to the remainder, we can continue dividing. If we candivide by LT (f1) or LT (f2), we proceed as usual, and if neither divides, we move the leadingterm of the intermediate dividend (the polynomial under radical) to the remainder column. Wecontinue dividing in such a way. Now the polynomial under the radical is y2 + y. It is notdivided by LT (f1) but we can divide it by LT (f2) leaving 1 and the subtract 1 · y2 from y2 + y.The obtained polynomial under the radical is y + 1 and neither LT (f1) nor LT (f2) dividesLT (y+1) = y. Therefore, we move y to the remainder column and obtain 1 which is also movedto the remainder. So, the remainder is x+ y + 1 and this concludes the example. Thus, we canwrite f in the form

x2y + xy2 + y2 = (x+ y) · (xy − 1) + 1 · (y2 − 1) + x+ y + 1.

Now one can ask whether the division algorithm has the same nice properties as in the one-variable version in k[x]. A first important property of the division algorithm in k[x] is that theremainder is uniquely determined. This fails when there is more than one variable. For example,in the previous case we have obtained the remainder r = x + y + 1. Now we change the orderof the divisors and we say f1 = y2 − 1 and f2 = xy − 1. Then the remainder is r = 2x + 1and we can see that the remainders are not the same and that the remainder is not uniquelycharacterized. The quotients ai and the remainder can change if we just rearrange the fi. Theyalso can change if we use another monomial ordering.

Using the division algorithm in k[x] one can solve the ideal membership problem, i.e. theproblem of whether or not the given polynomial f is the member of ideal I. The necessary and

sufficient condition in this case is that f ∈ I = ⟨h⟩ if and only if fh−→ 0. The question is if we

12 1 Introduction

have something similar in the case of a polynomial ring of more than one variable. Generally

speaking, the answer is “no”. Some examples show that fF−→ 0 is a sufficient condition but

not a necessary condition for a polynomial f(x1, . . . , xn) to be in the ideal generated by the setF , since we have seen that the remainder depends on the chosen monomial order and moreover,on the order of polynomials in the generating set F of the ideal I. However, it is reasonable toask whether there might be such a generating set for I for which the remainder r of divisionby these generators would be uniquely determined and the condition that the remainder is zerowould be equivalent to the membership in the ideal. As we will see, Grobner bases have theseproperties.

Fixing a term order > on k[x1, . . . , xn], any f ∈ k[x1, . . . , xn] may be written in the standardform, with respect to the order,

f = a1xα1 + a2x

α2 + · · · + asxαs , (1.14)

where αi = αj for i = j and 1 ≤ i, j ≤ s, and where, with respect to the specified term order,α1 > α2 > · · · > αs. The leading term LT (f) of f is the term LT (f) = a1x

α1 , the leadingmonomial LM(f) of f is the monomial LM(f) = xα1 and the leading coefficient LC(f) of f isthe coefficient LC(f) = a1.

Definition 1.2.5. A Grobner basis (also called a standard basis) of an ideal I ink[x1, . . . , xn] is a finite nonempty subset G = g1, . . . , gm of I\0 with the following property: for every nonzero f ∈ I, there exists gj ∈ G such that LT (gj)|LT (f).

As we see from the following statement using a Grobner basis we obtain the uniqueness ofthe remainder which was not assured when we divided by an arbitrary set of polynomials.

Proposition 1.2.6 ([8, 10]). Let G be a Grobner basis for a nonzero ideal I in k[x1, . . . , xn]and f ∈ k[x1, . . . , xn]. Then the remainder of f with respect to G is unique.

If I ⊂ k[x1, . . . , xn] is a nonzero ideal, G = g1, . . . , gs is a finite set of nonzero elements

of I, then G is a Grobner basis for I if and only if (∀f ∈ k[x1, . . . , xn] f ∈ I ⇔ fG−→ 0) (see

[23, 56]).We now describe an algorithm for computing a Grobner basis of a polynomial ring. For that

we recall few notions.Let f and g be from k[x1, . . . , xn] with LT (f) = axα and LT (g) = bxβ.The least common multiple of xα and xβ, denoted LCM(xα,xβ), is the monomial xγ =

xγ11 · · ·xγnn such that γj = max(αj , βj), 1 ≤ j ≤ n, and the S-polynomial of f and g is thepolynomial

S(f, g) =xγ

LT (f)f − xγ

LT (g)g.

Example 1.2.7. Let f1 = xy+1 and f2 = y+1 and fix lex with x > y. Then LCM(xy, y) = xy,and the S–polynomial is defined by

S(f1, f2) =xy

xy(xy + 1) − xy

y(y + 1) = −x+ 1.

By Buchberger’s Criterion (see [8]) if I is a nonzero ideal in k[x1, . . . , xn], < is a fixed termorder on k[x1, . . . , xn], then a generating set G = g1, . . . , gs is a Grobner basis for I with

respect to < if and only if S(gi, gj)G−→ 0 for all i = j.


Next, we present an algorithm for computing a Grobner basis of a polynomial ideal, whichis due to Buchberger [9].

Buchberger’s algorithm

Input: A set of polynomials G := f1, . . . , fs ∈ k[x1, . . . , xn].

Step 1: For each pair gi, gj ∈ G, i = j, compute the S-polynomial S(gi, gj) and compute theremainder rij of division S(gi, gj) by the set G.

Step 2: Check if all rij are equal to zero. If “yes”, then G is a Grobner basis, otherwise addall nonzero rij to G and return to step 1.

It is proved in [8] that the algorithm terminates and returns a Grobner basis of the idealI = ⟨f1, . . . , fs⟩.

Nowadays all major computer algebra systems (Mathematica, Maple, Reduce, Singu-lar, Macaulay and others) have routines to compute Grobner bases.

Even if a term order is fixed, an imprecision in the computation of a Grobner basis arisesbecause the division algorithm can produce different remainders for different orderings of poly-nomials in the set of divisors. Thus the output of Buchberger’s Algorithms is not unique. Also,usually the basis that it produces contains more polynomials than are necessary.

A Grobner basis G = g1, . . . , gm is called minimal if, for all i, j ∈ 1, . . . ,m, LC(gi) = 1and for j = i, LM(gi) does not divide LM(gj). Every nonzero polynomial ideal has a minimalGrobner basis (see [23, 56]). If no term of gi is divisible by any LT (gj) for j = i then the Grobnerbasis is called reduced and if we fix a term order then every nonzero ideal I ⊂ k[x1, . . . , xn] hasa unique reduced Grobner basis with respect to this order.

As we mentioned above knowing a Grobner basis we can immediately answer the IdealMembership Problem which asks whether or not a polynomial f is an element of a given idealI. A similar problem as the Ideal Membership Problem is the problem of equality of ideals. Itis easily proved that nonzero ideals I and J in k[x1, . . . , xn] are the same ideals if and only if Iand J have the same reduced Grobner basis with respect to a fixed term order.

Recall that to solve a system of linear equations, an effective method is to reduce the systemto an equivalent one in which an initial string of variables are missing from some of the equations,that is, to the so-called ”row-echelon” form. The next definition and theorem provide a way toeliminate a group of variables from a system of nonlinear polynomials. Moreover, it providesa way to find all solutions of a polynomial system in the case that the solution set is finite,or in other words, to find the variety of a polynomial ideal in the case that the variety iszero-dimensional.

Definition 1.2.8. Let I be an ideal in k[x1, . . . , xn] (with the implicit ordering of the variablesx1 > · · · > xn) and fix ℓ ∈ 0, 1, . . . , n − 1. The ℓ-th elimination ideal of I is the idealIℓ = I ∩ k[xℓ+1, . . . , xn].

Theorem 1.2.9 (Elimination Theorem). Fix the lexicographic term order on the ring k[x1, . . . , xn]with x1 > · · ·xn and let G be a Grobner basis for an ideal I of k[x1, . . . , xn] with respect to thisorder. Then for every ℓ, 0 ≤ ℓ ≤ n− 1, the set

Gℓ = G ∩ k[xℓ+1, . . . , xn]

is the Grobner basis for the ℓth elimination ideal Iℓ.

14 1 Introduction

The proof of the Elimination Theorem can be found in [23, 56].

Theorem 1.2.10 (Weak Hilbert Nullstellensatz [23]). If I is an ideal in C[x1, . . . , xn] such thatV(I) = ∅, then I = C[x1, . . . , xn].

The Weak Hilbert Nullstellensatz provides a way for checking whether or not a given poly-nomial system of the form (1.11) has a solution over C. By the theorem in order to find out ifthere is a solution to (1.11) it is sufficient to compute a reduced Grobner basis of ⟨f1, . . . , fs⟩using any term order. A polynomial system has no solution over C if and only if the reducedGrobner basis G for ⟨f1, . . . , fs⟩ with respect to any term order on C[x1, . . . , xn] is equal to 1.

If we are interested in solutions of (1.11) over a field k which is not algebraically closed, thena Grobner basis computation gives a definite answer only if it is equal to 1. For example, ifthere is no solution in Cn, then certainly there is no solution in Rn.

Let I ⊂ k[x1, . . . , xn] be an ideal. The radical of I, denoted√I, is the set

√I = f ∈ k[x1, . . . , xn] : there exists p ∈ N such that fp ∈ I.

An ideal J ∈ k[x1, . . . , xn] is called a radical ideal if J =√J .

If I is an ideal, then√I is also an ideal and determines the same affine variety as I:

V(√I) = V(I). (1.15)

The next theorem states that if a polynomial f vanishes at all points of some variety V(I) ⊂Cn, then some power of f must belong to I, that is, f belongs to the radical of I.

Theorem 1.2.11 (Strong Hilbert Nullstellensatz [23]). Let f, f1, . . . , fs ∈ C[x1, . . . , xn]. Thenf ∈ I(V(f1, . . . , fs)) if and only if there exists an integer m ≥ 1 such that fm ∈ ⟨f1, . . . , fs⟩. Inother words, for any ideal I ⊂ C[x1, . . . , xn],

√I = I(V(I)). (1.16)

This theorem justifies the so-called radical membership test. The test says that for a polyno-mial f and an ideal I = ⟨f1, . . . , fm⟩ in k[x1, . . . , xn], f ∈

√I if and only if the reduced Grobner

basis of the ideal ⟨1 −wf, f1, . . . , fm⟩ (here w is a new variable) is equal to 1. Geometrically,that means that the polynomial f is zero on the variety V(I).

The Grobner basis theory allows one to find all solutions of system (1.11) in the case thesystem has only finite number of solutions. In such case a Grobner basis with respect to thelexicographic order is always in a “row-echelon” form, as can be seen using the following example.Consider the polynomials

f1 = 8x2y2 + 5xy3 + 3x3z + x2yz,

f2 = x5 + 2y3z2 + 13y2z3 + 5yz4, (1.17)

f3 = 8x3 + 12y3 + xz2 + 3,

f4 = 7x2y4 + 18xy3z2 + y3z3.

With respect to the lexicographic order with x > y > z, the reduced Grobner basis for theideal generated by f1, f2, f3, f4 in Q[x, y, z] is g1 = x, g2 = y3 + 1

4 , g3 = z2. Thus, the systemf1 = f2 = f3 = f4 = 0 is equivalent to the system

x = 0, y3 + 14 = 0, z2 = 0.


For a generic system a Grobner basis has a much more complicated structure than in thisexample. However, if the system has only finite number of solutions (that is, the ideal iszero-dimensional), then any reduced Grobner basis g1, . . . , gt with respect to lexicographicorder must contain a polynomial in one variable, let say, g1(x1). Then, there is a groupof polynomials in the Grobner basis depending on this variable and one more variable, say,g2(x1, x2), . . . , gt(x1, x2) etc. Thus, we first solve (perhaps only numerically) the equationg1(x1) = 0. Then, for each solution x∗1 of g1(x1) = 0 we find the solutions of g2(x

∗1, x2) =

· · · = gt(x∗1, x2) = 0, which is a system of polynomials in a single variable x2. Continuing the

process we obtain in this way all solutions of our system (1.11). Thus, in the case of the finitenumber of solutions theoretically Grobner bases computations provide the complete solution tothe problem (see e.g. Section 2.2. of [1] for more details).

1.2.2 Operations on ideals and varieties

The situation in which the variety of a polynomial ideal consists of finite number of points arisesrarely. In the generic case the variety consists of infinitely many points, so, generally speaking“to solve” a polynomial system means to find a decomposition of the variety of the ideal intoirreducible components, which we will discuss after we define two ideal operations needed tocompute decompositions of ideals. The first operation is the intersection of two ideals. Theintersection of I and J is the set

I ∩ J = f ∈ k[x1, . . . , xn] : f ∈ I and f ∈ J.

It is easy to see that the union of varieties is equal to the variety of the intersection of ideals:

V(I) ∪V(J) = V(I ∩ J).

In the next proposition we give an algorithm to compute a Grobner basis of the intersection oftwo ideals.

Proposition 1.2.12 ([23, 56]). Let I = ⟨f1, . . . , fu⟩ and J = ⟨g1, . . . , gv⟩ be ideals in k[x1, . . . , xn].We form an ideal G

′= ⟨ tf1(x), . . . , tfu(x), (1 − t)g1(x), . . . , (1 − t)gv(x) ⟩ ⊂ k[t, x1, . . . , xn] and

compute a Grobner basis G of the ideal G′with respect to lexicographic order with t > x1 >

· · · > xn. Then G ∩ k[x1, . . . , xn] is a Grobner basis for I ∩ J .

Next, in order to study the set theoretic difference of varieties we need to introduce the notionof the quotient of two ideals. If I and J are ideals in k[x1, . . . , xn] then their ideal quotient I : Jis the ideal

I : J = f ∈ k[x1, . . . , xn] : fg ∈ I for all g ∈ J.

Whether or not a set S ⊂ kn is an affine variety, the set

I(S) = f ∈ k[x1, . . . , xn] : f(a) = 0 for all a ∈ S

is an ideal in k[x1, . . . , xn]. It is easy to see that it is a radical ideal.

Definition 1.2.13. The Zariski closure of a subset S of an affine space kn is the smallest affinealgebraic variety containing S.

16 1 Introduction

If S ∈ kn, the Zariski closure of S is denoted by S and is equal to V(I(S)). The next theoremshows how the quotient of ideals is related to the set theoretical difference of their varieties. Aproof can be found in [23].

Theorem 1.2.14. Let I and J be ideals in k[x1, . . . , xn]. Then

V(I : J) ⊃ V(I)\V(J).

If, in addition, k is algebraically closed and I is a radical ideal, then

V(I : J) = V(I)\V(J).

The quotient of two ideals can be computed using the following proposition, whose proofcan be found in [23, 56].

Proposition 1.2.15. Let I and J1, . . . , Jm be ideals in k[x1, . . . , xn] and g a nonzero elementof k[x1, . . . , xn]. If h1, . . . , hs is a basis for I ∩ ⟨g⟩, then h1/g, . . . , hs/g is a basis for I : ⟨g⟩and

I : (m∑s=1

Js) = ∩ms=1(I : Js).

An affine variety V ⊂ kn is irreducible if, whenever V = V1 ∪ V2 for affine varieties V1 andV2, either V1 = V or V2 = V . Let I be an ideal and V = V(I) its variety. It is known that anyaffine variety can be decomposed into the irreducible varieties, that is, it can be written as afinite union

V = V1 ∪ . . . ∪ Vm,

where each Vi is an irreducible variety.A decomposition

V = V1 ∪ . . . ∪ Vm,

in which each Vi is an irreducible variety is called a minimal decomposition if Vi * Vj for i = j.By Theorem 4 in [23, §4, Chapter 4] if V is an affine variety then V has a minimal decomposition.

We recall that an ideal I ⊂ k[x1, . . . , xn] is prime if whenever f, g ∈ k[x1, . . . , xn] and fg ∈ I, then either f ∈ I or g ∈ I.

The radical of I is an intersection of prime ideals,√I = ∩s

j=1Pj . For example, J =√J =

⟨xy, xz⟩ = ⟨x⟩ ∩ ⟨y, z⟩, that is, the variety of J is the union of the plane x = 0 and the liney = z = 0.

As is well known, every positive integer is a product of prime numbers. This implies thatany ideal in the ring Z is an intersection of prime ideals. However, such property does not holdfor ideals in k[x1, . . . , xn] – we can not write an arbitrary ideal in k[x1, . . . , xn] as an intersectionof prime ideals.

Definition 1.2.16. An ideal I ∈ k[x1, . . . , xn] is primary if fg ∈ I implies either f ∈ I or somepower gm ∈ I (for some m > 0).

While every radical ideal can be written as an intersection of prime ideals, an arbitrary idealI can be written by the Lasker-Noether theorem (see [23]) as an intersection of primary ideals,

I = ∩sj=1Qj . (1.18)


We say that (1.18) defines a primary decomposition of the ideal I. Ideals Pi =√Qi are called

minimal associate primes of I.For computing primary decompositions in the computer algebra system Singular we can

use two routines, primdecGTZ or primdecSY [25] based on the Gianni-Trager-Zacharias algorithm(see [30]) and on the Shimoyama-Yokoyama algorithm (see [64]), respectively.

A variety V can sometimes be described using parametric equations. One difficulty which weface here is that the parametrization might not cover all of the variety V . The so-called implicit-ization problem asks for the equations defining the smallest variety containing the parametrizedset.

In the next theorem we see what happens when we have a parametrization by rationalfunctions. The proof can be found in [23].

Theorem 1.2.17 (Rational Implicitization). Let k be an infinite field, F : km\W → kn, whereW = V(g1g2 · · · gn), be the function determined by the rational parametrization

x1 =f1(t1, . . . , tm)

g1(t1, . . . , tm),

...

xn =fn(t1, . . . , tm)

gn(t1, . . . , tm),

(1.19)

where f1, g1, . . . , fn, gn are polynomials in k[t1, . . . , tn]. Let J be the ideal J = ⟨g1x1−f1, . . . , gnxn−fn, 1 − gy⟩ ⊂ k[y, t1, . . . , tm, x1, . . . , xn], where g = g1g2 · · · gn and let Jm+1 = J ∩K[x1, . . . , xn]be the (m + 1)-th elimination ideal. Then V(Jm+1) is the smallest variety in kn containingF (km\W ).

Theorem 1.2.17 implies the following implicitization algorithm for rational parametrizations:if we have equations (1.19) with polynomials f1, g1, . . . , fn, gn, introduce a new variable y andthe ideal J = ⟨g1x1 − f1, . . . , gnxn − fn, 1 − gy⟩, where g = g1g2 · · · gn. Compute a Grobnerbasis with a lexicographic order with y > ti : i = 1, . . . ,m > xi : i = 1, . . . , n and takeonly polynomials depending on xi. Then these polynomials define the smallest variety in kn

containing the parametrized set.

1.2.3 Decomposition of varieties using modular arithmetics

Unfortunately, in practise, calculations of Grobner bases, especially with respect to lexicographicorders, very often face tremendous computational difficulties. During the execution of algorithmsthe size of coefficients of the S-polynomials grows exponentially. Even for the very simpleexample of the polynomials considered above

f1 = 8x2y2 + 5xy3 + 3x3z + x2yz,

f2 = x5 + 2y3z2 + 13y2z3 + 5yz4,

f3 = 8x3 + 12y3 + xz2 + 3,

f4 = 7x2y4 + 18xy3z2 + y3z3,

with the reduced Grobner basis of the ideal ⟨f1, f2, f3, f4⟩, G = g1, g2, g3, where

g1 = x, g2 = y3 +1

4, g3 = z2,

18 1 Introduction

the following polynomial appears in the intermediate computations of the Grobner basis:

y3 − 1735906504290451290764747182 · · · . (1.20)

The integer in the second term of the above polynomial contains roughly 80,000 digits [3]. It isthe numerator of a rational number with roughly an equal number of digits in the denominator.

This notorious computational difficulty of the Grobner basis calculations over the field ofrational numbers is an essential obstacle for using the Grobner basis theory for real worldapplications. We now describe an approach relying on modular arithmetic which can be appliedin order to drastically simplify finding the set of solutions of a polynomial system.

To perform modular computations one chooses a prime number p and does all calculationsmodulo p, that is, in the finite field of the characteristic p (the field Zp = Z/p). It turns outthat the modular calculations still keep essential information on the original system and it isoften possible to extract this information from the result of calculations in Zp and to obtain theexact solution of the polynomial system over the field of rational numbers.

To perform the rational reconstruction, that is, to reconstruct r/s ∈ Q given its image t ∈ Zp,we use the following algorithm of [67] (in the algorithm ⌊·⌋ stands for the floor function).

Rational reconstruction algorithm

Step 1. u = (u1, u2, u3) := (1, 0,m), v = (v1, v2, v3) := (0, 1, c)Step 2. While

√m/2 ≤ v3 do q := ⌊u3/v3⌋, r := u− qv, u := v, v := r

Step 3. If |v2|≥√m/2 then error()

Step 4. Return v3, v2

Given an integer c and a prime number p the algorithm produces integers v3 and v2 such thatv3/v2 ≡ c (mod p), that is, v3 = v2c+pt with some t. If such a number v3/v2 does not exist, thenthe algorithm returns ”error()”. A Mathematica code to perform the rational reconstructionalgorithm is given in Appendix A.

For example, computing the Grobner basis of (1.17) over the field of characteristic 32003 wefind G = x, y3 +8001, z2. Rational reconstruction yields 8001 ≡ 1/4 (mod 32003). Thereforethe reconstructed (lifted) Grobner basis is G = x, y3 + 1/4, z2. Now no long numbers (likein (1.20)) appear in the intermediate polynomials, all numbers have at most 5 digits, thereforethe speed of calculation essentially increases and the memory consumption falls drastically.

We now can describe an approach to solving large polynomial systems (1.11), suggestedin [55]. It is called the Decomposition Algorithm with Modular Arithmetics and consists offollowing five steps.

• Choose a prime number p and compute the minimal associated primes Q1, . . . , Qs of I inZp[x1, . . . , xn].

• Using the rational reconstruction algorithm lift the ideals Qi (i = 1, . . . , s) to the ideals Qi

in Q[x1, . . . , xn] (that is, replace all coefficients of Qi by the rational numbers computedwith the reconstruction algorithm).

• For each i = 1, . . . , s using the radical membership test check whether the polynomialsf1, . . . , fs are in the radicals of the ideals Qi, that is, whether the reduced Grobner basisof the ideal ⟨1 − wf,Qi⟩ is equal to 1. If ”yes”, then go to the step 4, otherwise takeanother prime p and go to step 1.


• Compute Q = ∩si=1Qi ⊂ Q[x1, . . . , xn].

• Check that√Q =

√I, that is, that for any g ∈ Q the reduced Grobner basis of the ideal

⟨1 − wg, I⟩ is equal to 1 and for any f ∈ I the reduced Grobner basis of the ideal⟨1 − wf,Q⟩ is equal to 1. If it is the case then V(I) = ∪s

i=1V(Qi). If not, then chooseanother prime p and go to Step 1.

In the following sections we will apply this algorithm to solve large polynomial systemsarising in our study of the integrability problem.

Chapter 2

The problems of center and localintegrability

An essential part of the theory of integrability of ODE’s is devoted to studying local integrabilityof two dimensional analytic systems of differential equations (two dimensional analytic vectorfields) in a neighborhood of a singular point of center or focus type. Most works on the subjectare devoted to investigation of polynomial vector fields.

The classification of polynomial systems having a center leads to systems which are notintegrable by means of known mechanisms and therefore requires the development of new meth-ods for investigating integrability. Thus, such studies stimulate developing new methods andapproaches for finding new mechanisms of integrability in polynomial differential systems.

Such studies can be also considered as a starting point for investigation of limit cycle bifur-cations in polynomial differential systems. As is known, interesting and rich bifurcations occurin a neighborhood of the center variety, that is, when we perturb systems having a center at theorigin. Thus, the next important step after classification of centers of a polynomial family is theinvestigation of limit cycle bifurcations under small change of parameters in a neighborhood ofthe center variety. This is a fascinating problem called the cyclicity problem.

In this chapter we first describe a general approach to studying the center and integrabilityproblems and then we focus our attention on the investigation of these problems for the followingpolynomial families:

x = x(1 − a10x− a20x2 − a11xy − a02y

2),

y = −y(1 − b01y − b02y2 − b11xy − b20x

2),

x = x(1 − a30x3 − a21x

2y − a12xy2 − a03y

3),y = −y(1 − b30x

3 − b21x2y − b12xy

2 − b03y3),

andx = x− a40x

5 − a31x4y − a22x

3y2 − a13x2y3 − a04xy

4 − y5,

y = −y + x5 + b40x4y + b31x

3y2 + b22x2y3 + b13xy

4 + b04y5.

21

22 2 The problems of center and local integrability

2.1 The Poincare first return map and the center problem

Poincare and Lyapunov [50, 40] proved that the elementary point at the origin of the system ofdifferential equations

du

dt= αu− βv +

∞∑i+j=2

αijuivj = αu− βv + P (u, v),

dv

dt= βu+ αv +

∞∑i+j=2

βijuivj = βu+ αv + Q(u, v),

(2.1)

where α,β, αij , βij ∈ R, is a center if and only if α = 0 and the system admits a first integral ofthe form

Φ = u2 + v2 +∑

k+l≥3

ϕklukvl. (2.2)

This is the so-called Poincare-Lyapunov Theorem (see Theorem 2.2.2 in the next section). Thetheorem says that the qualitative picture of trajectories in a neighborhood of the singular pointis related to local integrability of the system: the singular point is a center if and only if thereexists an analytic first integral of the form (2.2). It can be also proved that there exists ananalytic first integral of system (2.1) if and only if there exists a formal first integral of system(2.1) of the form (2.2). However, the Poincare-Lyapunov Theorem does not give an answer tothe question how to establish whether for a given system of differential equations there exists afirst integral of the form (2.2). The answer to this question must be found for each particularsystem separately, and so far we have no general method that enables us to answer this questionfor an arbitrary system (2.1). Thus, it remains an open problem how to identify systems witha center within a given parametric family of planar polynomial systems of ordinary differentialequations. This is the so-called problem of distinguishing between a center and a focus, or thePoincare center problem.

Although the problem of distinguishing between a center and a focus has been studied inmany works it is solved only for quadratic systems [26], for the systems in the form of a linearcenter perturbed with homogeneous cubic nonlinearities [45], for the so-called Kukles system[43, 63], for some linear centers perturbed with homogeneous polynomials of degree five [13],and for a few other specific families of polynomial systems of ODE’s. Even the case of thegeneral cubic system (that is, system of the form (2.1), where the expressions on the right-handside are cubic polynomials) is still unsolved.

One of the tools to study the problem of distinguishing between a center and a focus is thePoincare return map, which can be computed as follows.

Let P (u, v) =∑∞

k=2 P(k)(u, v) and Q(u, v) =

∑∞k=2Q

(k)(u, v), where P (k)(u, v) andQ(k)(u, v)are homogeneous polynomials of degree k.

Introducing polar coordinates x = r cosφ, y = r sinφ, we obtain

r =αr + P (r cosφ, r sinφ) cosφ+ Q(r cosφ, r sinφ) sinφ

φ =β − r−1[P (r cosφ, r sinφ) sinφ− Q(r cosφ, r sinφ) cosφ].(2.3)

Dividing the first equation in (2.3) by the second one, we obtain

dr

dφ=αr + r2F (r, sinφ, cosφ)

β + rG(r, sinφ, cosφ)= R(r, φ). (2.4)

2.1 The Poincare first return map and the center problem 23

The function R(r, φ) is a 2π-periodic function of φ and is analytic for all φ and for |r|< r∗,for some sufficiently small r∗. The fact that the origin is a singularity for (2.1) corresponds tothe fact that R(0, φ) ≡ 0, so that r = 0 is a solution of (2.4). We can expand R(r, φ) in a powerseries in r:

dr

dφ= R(r, φ) = rR1(φ) + r2R2(φ) + · · · =

α

βr + · · · , (2.5)

where Rk(φ) are 2π-periodic functions of φ. The series is convergent for all φ and for allsufficiently small r.

Denote by r = f(φ,φ0, r0) the solution of (2.5) with initial conditions r = r0 and φ = φ0.The function f(φ,φ0, r0) is an analytic function of all three variables φ,φ0 and r0 and has theproperty that

f(φ,φ0, 0) ≡ 0 (2.6)

(because r = 0 is a solution of (2.5)). From equation (2.6) using continuous dependence ofsolutions on parameters we conclude that every trajectory of system (2.3) in a sufficiently smallneighborhood of the origin crosses every ray φ = c, 0 ≤ c < 2π. This implies that in orderto investigate the trajectories in a sufficiently small neighborhood of the origin it is sufficientto consider all trajectories passing through a segment Σ = (u, v) : v = 0, 0 ≤ u ≤ r∗ forr∗ sufficiently small, that is, all solutions r = f(φ, 0, r0). The function r = f(φ, 0, r0) can beexpanded in a power series in r0,

r = f(φ, 0, r0) = w1(φ)r0 + w2(φ)r20 + · · · , (2.7)

which is convergent for all 0 ≤ φ ≤ 2π and for |r0|< r∗. This function is a solution of (2.5),hence

w′1r0 + w

′2r

20 + · · ·

≡ R1(φ)(w1(φ)r0 + w2(φ)r20 + · · ·) +R2(φ)(w1(φ)r0 + w2(φ)r20 + · · ·)2 + · · · ,

where the derivatives are taken with respect to the variable φ. Equating the coefficients of likepowers of r0 in this identity, we obtain the recurrence differential equations

w′1 = R1(φ)w1,

w′2 = R1(φ)w2 +R2(φ)w2

1,

w′3 = R1(φ)w3 + 2R2(φ)w1w2 +R3(φ)w3

1,

...

(2.8)

Taking into account the initial condition r = f(0, 0, r0) = r0 we obtain

w1(0) = 1, wj(0) = 0 for j > 1. (2.9)

Using these conditions we can consequently find the functions wj(φ) by integrating equations(2.8). In particular,

w1(φ) = eαβφ. (2.10)

Setting φ = 2π in the solution r = f(φ, 0, r0) we obtain the value r = f(2π, 0, r0), correspondingto the point of Σ where the trajectory r = f(φ, 0, r0) first intersects Σ again.

We can now define few a fundamental concepts of the qualitative theory of planar systemsof ODE’s.


Definition 2.1.1. Fix the system of the form (2.1).

(a) The functionR(r0) = f(2π, 0, r0) = η1r0 + η2r

20 + η3r

30 + · · · (2.11)

(defined for |r0|< r∗), where η1 = w1(2π) and ηj = wj(2π) for j ≥ 2, is called the Poincarefirst return map (or simply the return map).

(b) The functionP(r0) = R(r0) − r0 = η1r0 + η2r

20 + η3r

30 + · · · (2.12)

is called the difference function.

(c) The coefficient ηj, j ∈ N, is called the jth Lyapunov number.

Zeros of (2.12) correspond to cycles (closed orbits, that is, orbits that are ovals) of system(2.1); isolated zeros correspond to limit cycles (isolated closed orbits). It is not difficult to provethat the first nonzero coefficient of the expansion (2.12) is the coefficient of an odd power of r0.

The Lyapunov numbers completely determine the behavior of the trajectories of system (2.1)near the origin.

Theorem 2.1.2. System (2.1) has a center at the origin if and only if all Lyapunov numbersare zero. Moreover, if η1 = 0, or if for some k ∈ N

η1 = η2 = · · · = η2k = 0, η2k+1 = 0, (2.13)

then all trajectories in a neighborhood of the origin are spirals and the origin is a focus, whichis stable if η1 < 0 or (2.13) holds for η2k+1 < 0 and is unstable if η1 > 0 or (2.13) holds forη2k+1 > 0.

A proof can be found in [56]. From (2.10) and the theorem we see that if α < 0, then theorigin of (2.1) is stable focus and if α > 0 then the origin is unstable focus.

2.2 Complexification of the system, the center variety and focus quantities 25

2.2 Complexification of the system, the center variety and focusquantities

One of the main difficulties in the study of the center problem comes from the complexity incomputing the irreducible decomposition of the variety of the ideal generated by the Lyapunovquantities that are the coefficients of the Poincare first return map. Since it is easier to studycomplex varieties than real ones we complexify the real system as follows. We will assume thatin (2.1) α = 0 (since otherwise the system has a focus at the origin) and the functions P and Qare polynomials of degree n. First setting x = u+ iv system (2.1) becomes the equation

x = R(x, x).

Adjoining to this equation its complex conjugate we have the system

x = R(x, x), x = R(x, x).

Consider y := x as a new variable and R as a new function. Then, from the latter system weobtain a system of two complex differential equations which we can write in the form

x = i(x−n−1∑

p+q=1

ap,qxp+1yq) = P (x, y)

y = −i(y −n−1∑

p+q=1

bq,pxqyp+1) = Q(x, y).

(2.14)

After the change of time idt = dτ and rewriting t instead of τ , system (2.14) becomes

x = x−n−1∑

p+q=1

ap,qxp+1yq = x− P (x, y)

y = −y +

n−1∑p+q=1

bq,pxqyp+1 = −y +Q(x, y).

(2.15)

Here p ≥ −1 and q ≥ 0 and we denote the vector of coefficients of system (2.15) by (a, b) =(a1,0, a0,1, . . . , a−1,n, b1,0, b0,1, . . . , bn,−1). System (2.14) (and also (2.15)) is called the complexi-fication of system (2.1).

If in (2.14) y = x and bqp = apq, then system (2.14) has a real preimage. However we willallow in our considerations the full set of systems of the form (2.14) without the requirementthat bqp = apq. Thus throughout this thesis we take C2ℓ (ℓ is the cardinality of the index set(p, q) : 1 ≤ p+ q ≤ n− 1, p ∈ N∪−1, 0, q ∈ N∪0) as the parameter space of (2.14), andwill denote it by E(a, b). E(a) = E(a, a) will denote the parameter space of

x = P (x, x) = i(x−n−1∑

p+q=1

apqxp+1xq), (2.16)

which we call the real polynomial system in complex form. To simplify the notation we willdenote by C[a, b] the polynomial ring in the variables apq, bqp over C. When the context makes


it clear that we are considering systems of the form (2.14), for economy of expression we willspeak of “the system (a, b) ” when we mean the system of the form (2.14) or (2.15) with thechoice (a, b) of parameter values.

We introduce also the notion of normal forms of systems of ODE’s which are important forstudies of local integrability. They will play also an important role in computing the isochronicityquantities in the last chapter of this thesis.

To this end, we consider the system

x = Ax + X(x), (2.17)

where x ∈ Cn, A is a possibly complex n×n matrix, and each component Xk(x) of X, 1 ≤ k ≤ nis a formal or convergent power series, that contains no constant or linear terms.

We say that system (2.17) is formally equivalent to a system

y = Ax + Y(y), (2.18)

if there is a change of variablesx = H(y) = y + h(y) (2.19)

that transforms (2.17) into (2.18), where the coordinate functions of Y and h are formal powerseries, where

Y = (Y1(y), . . . , Yn(y))T , h = (h1(y), . . . , hn(y))T

andYm =

∑|α|≥2

Y (α)m yα, hm =

∑|α|≥2

h(α)m yα, m = 1, . . . , n.

Let κ1, . . . , κn be the eigenvalues of the matrix A in (2.17), ordered according to the choiceof a Jordan normal form J of A, and let κ = (κ1, . . . , κn). Suppose m ∈ 1, . . . , n and α ∈ Nn

0 ,|α|= α1 + · · · + αn ≥ 2, are such that

(α, κ) − κm = 0.

Then m and α are called a resonant pair, the corresponding coefficient X(α)m of the monomial

xα in the mth component of X is called a resonant coefficient, and the corresponding term iscalled a resonant term of X. The coefficients that are not resonant are called nonresonant.

Definition 2.2.1. A normal form for system (2.17) is a system

x = Jx + X(x),

in which every nonresonant coefficient is equal to zero. A normalizing transformation for system(2.17) is any (possibly formal) change of variables (2.19) that transforms (2.17) into a normalform. A normalizing transformation is called distinguished if for each resonant pair m and α,the corresponding coefficient hαm is zero. A normal form obtained by means of a distinguishednormalizing transformation is called a distinguished normal form.

By Propositin 3.2.1 in [56] if we fix real system (2.1) for which α = 0 and β = 0 then thereis a normal form of its complexification (2.14), which is produced by the unique distinguished

transformation, and whose coefficients satisfy Y(j,k)1 = Y

(k,j)2 for all j + k ≥ 2.

2.2 Complexification of the system, the center variety and focus quantities 27

LetG = Y1 + Y2, H = Y1 − Y2, (2.20)

which we can expand in

G(w) =∞∑k=1

G2k+1wk, H(w) =

∞∑k=1

H2k+1wk, (2.21)

where w = y1y2.We recall that a first integral on an open set Ω in R2 or C2 of a smooth or analytic system

of differential equationsx1 = f1(x1, x2), x2 = f2(x1, x2) (2.22)

defined everywhere on Ω is a nonconstant differentiable function Ψ : Ω → C that is constant ontrajectories. A formal first integral is a formal power series Ψ(x1, x2) in x1, x2, not all of whosecoefficients are zero, which under term-by-term differentiating satisfies ∂Ψ

∂x1f1 + ∂Ψ

∂x2f2 ≡ 0 on Ω.

For the proof of the next theorem, which is a version of the Poincare-Lyapunov theorem, onecan see [56].

Theorem 2.2.2. The following statements about system (2.1) are equivalent:

(a) system (2.1) has a center at the origin;

(b) the Lyapunov numbers η2k+1 are all zero;

(c) system (2.1) has a formal first integral of the form Φ = u2 + v2 + · · ·;

(d) the complexification (2.14) of system (2.1) has a formal first integral of the form Ψ = xy+· · ·

(e) the coefficients G2k+1 of the function (2.20) computed from the normal form of the com-plexification (2.14) of (2.1) are all zero.

Following Dulac [26], it is common to use statement (d) of the theorem to extend the conceptof a center to certain complex systems.

Definition 2.2.3. Consider the system

x = x− P (x, y)

y = −y +Q(x, y),(2.23)

where x and y are complex variables and P and Q are complex series without constant or linearterms that are convergent in a neighborhood of the origin. We say that system (2.23) has acenter at the origin if it has a formal first integral of the form

Ψ(x, y) = xy +∑

j+k≥3

vj−1,k−1xjyk. (2.24)

If in (2.23) P and Q satisfy the condition Q(x, x) = P (x, x), then the system

x = i(x− P (x, y))

y = −i(y −Q(x, y)),(2.25)


is the complexification of a real system. If there exists a formal first integral of the form (2.24)for system (2.25), then by Definition 2.2.3 this system has a center at the origin of C2 and, byTheorem 2.2.2, the corresponding real system (2.1) has a center at (0, 0) ∈ R2.

By the definition of first integral we have

DΨ :=∂Ψ

∂xP (x, y) +

∂Ψ

∂yQ(x, y) ≡ 0. (2.26)

To find necessary conditions for the existence of a first integral of the form (2.24) for system(2.14) we look for a formal series (2.24) satisfying (2.26). To start the computational processfor finding the first N conditions for integrability we write down the initial string of (2.24) upto order 2N + 1

Ψ2N+1(x, y) = xy +2N+1∑j+k=3

vj−1,k−1xjyk.

Then for each i = 3, . . . , 2N + 1 we equate coefficients of terms of order i in the expression

∂Ψ2N+1

∂xP (x, y) +

∂Ψ2N+1

∂yQ(x, y) (2.27)

to zero obtaining 2N − 2 systems of linear equations of unknown variables vjk. Then, we lookfor solutions of the linear systems obtained starting from system that corresponds to i = 3.Linear systems corresponding to odd i = 2ℓ − 1 always have unique solutions. After solvingthe system we substitute the obtained values of vjk into the linear systems corresponding toi > 2ℓ − 1. For systems that correspond to even i = 2ℓ, there is one equation more than thenumber of variables. After dropping a suitable equation one obtains the system with the uniquesolution. After solving the system we assign the value 0 for the undefined vjk (with j + k = 2ℓ)and substitute the obtained values of vjk into the linear systems corresponding to i > 2ℓ. Next,we evaluate (2.27) with the found vjk (j + k ≤ 2ℓ) and find the coefficient of xℓyℓ which wedenote by gℓ−1ℓ−1. Computing in this way we obtain a list of polynomials g11, g22, g33, . . . in theparameters of system (2.23).

It is evident that if for a fixed system with the coefficients (a∗, b∗), gkk(a∗, b∗) = 0 for allk ∈ N, then we can obtain the formal first integral Ψ since the process of constructing succeedsat every step. But the converse is not immediately apparent. Although in the general case afirst integral of the form (2.24) does not always exist, the construction process always yields aseries of the form (2.24) for which DΨ = ΨxP + ΨyQ reduces to

DΨ = g11(xy)2 + g22(xy)3 + g33(xy)4 + · · · . (2.28)

Definition 2.2.4 ([56]). The polynomial gkk ∈ C[a, b] on the right-hand side of (2.28) is calledthe kth focus quantity for the singularity at the origin of system (2.14). The ideal of focusquantities , B = ⟨g11, g22, . . . , gjj , . . .⟩ ⊂ C[a, b] is called the Bautin ideal, and the affine varietyVC = V(B) is called the center variety for the singularity at the origin of system (2.14).

We will denote by Bk the ideal generated by the first k focus quantities, Bk = ⟨g11, g22, . . . , gkk⟩.

2.3 Time-reversible and Hamiltonian systems, Darboux method 29

2.3 Time-reversible and Hamiltonian systems, Darboux method

In this section we discuss three mechanisms for proving the integrability of polynomial systems.First, we describe an approach to find all time-reversible systems inside of a family of polynomialsystems. Some main definitions and notions will be used also in the third chapter of this thesisfor studying limit cycle bifurcations. Then we mention briefly Hamiltonian systems. In thelast part of this section we discuss the third mechanism providing local integrability, Darbouxintegrability.

2.3.1 Time-reversibility

Time-reversible symmetry arises frequently in many studies in classical and quantum mechanics.We discuss systems having a symmetry that in the real case forces any singularity of focus orcenter type to be in fact a center.

The systemdz

dt= F (z),

where F : C2 → C2 (or F : R2 → R2), is time-reversible if there exists a linear transformationT : C2 → C2, such that

d(Tz)

dt= −F (Tz).

We will consider symmetries of the form

T : x→ γy, y = γ−1x (2.29)

with γ ∈ C (or R).We say that a line L is an axis of symmetry of system (2.30) if as point sets the orbits of the

system are symmetric with respect to L. When direction of flow is taken into account, there aretwo types of symmetry of a real system with respect to a line L: mirror symmetry, meaning thatwhen the phase portrait is reflected in the line L it is unchanged; and time-reversible symmetry,meaning that when the phase portrait is reflected in the line L and then the sense of every orbitis reversed (corresponding to a reversal of time), the original phase portrait is obtained.

Since it is easier to study the center problem for complex systems we first describe thegeneralization of this geometric time-reversibility to complex systems. We complexify the realsystem

u = U(u, v), v = V (u, v) (2.30)

by making use of the substitution x = u+ iv. Then, it is transformed into

x = P (x, x). (2.31)

As above, adding to this equation its complex conjugate and considering x as a new variabley and allowing the parameters of the second equation to be arbitrary, we obtain the complexi-fication of system (2.30),

x = P (x, y), y = Q(x, y). (2.32)

Simple calculations show (see e.g. [56] for more details) that system (2.32) is time-reversiblewith respect to a transformation (2.29) if and only if for some γ

γQ(γy, x/γ) = −P (x, y), γQ(x, y) = −P (γy, x/γ). (2.33)


Since we restrict our study to the case when P (x, y) and Q(x, y) in (2.32) are polynomials,without the lose of generality we can write such system in the form

x = −n−1∑

p+q=1

ap−1,qxpyq = P (x, y),

y =

n−1∑p+q=1

bq−1,pxqyp = Q(x, y).

(2.34)

We denote by (a, b) = (ap1,q1 , ap2,q2 , . . . , apℓ,qℓ , bqℓ,pℓ , . . . , bq2,p2 , bq1,p1) the ordered vector of coeffi-cients of system (2.34), by E(a, b) = C2ℓ the parameter space of (2.34) and by C[a, b] polynomialring in the variables apq in bqp over the field C. The condition (2.33) yields

bqp = γp−qapq apq = bqpγq−p. (2.35)

We rewrite (2.35) in the form

apkqk = tk, bqkpk = γpk−qktk (2.36)

for k = 1, . . . , ℓ. From a geometrical point of view equations (2.36) define a surface in theaffine space C3ℓ+1 = (ap1,q1 , ap2,q2 , . . . , apℓ,qℓ , bqℓ,pℓ , . . . , bq2,p2 , bq1,p1 , t1, . . . , tℓ, γ). Thus the set ofall time-reversible systems is the projection of this surface onto C2ℓ = E(a, b). Let

H = ⟨1 − wγ1 . . . γℓ, apkqk − tk, γkbqkpk − ˜γktk | k = 1, . . . , ℓ⟩, (2.37)

where γk = γqk−pk , ˜γk = 1 if pk − qk ≤ 0, γk = 1, ˜γk = γpk−qk if pk − qk > 0. Then by Theorem1.2.17 the minimal variety which contains the set of all time-reversible systems in the family(2.34) is

R = V(I) where I = C[a, b] ∩H (see [57] for more details). (2.38)

We see that the Zariski closure of the set of all time-reversible systems is the variety of the idealI. By Theorem 1.2.17 in order to find a generating set for the ideal I it is sufficient to computea Grobner basis for H with respect to an elimination order with w, γ, tk > apkqk , bqkpk | k =1, . . . , ℓ and take from the output list those polynomials which depend only on apkqk , bqkpk ,k = 1, . . . , ℓ.

As we have seen above, a real polynomial system that has a singularity which is either acenter or focus and which is time-reversible with respect to the line passing through this pointmust have a center at the point. In the next theorem (see [56] for a proof) it is stated that theanalogous fact for the complex systems of the form (2.14) is also true.

Theorem 2.3.1. Every time-reversible system of the form (2.14) (not necessarily satisfyingbqp = apq) has a center at the origin.

The set of time-reversible systems is not generally itself a variety, since it is defined by arational parametrization. For ν = (ν1, . . . , ν2ℓ) ∈ N2ℓ

0 let [ν] denote the monomial in C[a, b] givenby

[ν] = aν1p1,q1 · · · aνℓpℓ,qℓ

bνℓ+1qℓ,pℓ · · · bν2ℓq1,p1 , (2.39)


and ν denote the involution of ν, ν = (ν2ℓ, . . . , ν1). For a fixed family (2.34) and the correspond-ing mapping L given by

L(ν) =

(p1q1

)ν1 + · · · +

(pℓqℓ

)νℓ +

(qℓpℓ

)νℓ+1 + · · · +

(q1p1

)ν2ℓ, (2.40)

defineM = ν ∈ N2ℓ

0 : L(ν) = (j, j) for some j ∈ N0 (2.41)

and let Isym be the ideal defined by

Isym = ⟨[ν] − [ν] : ν ∈ M ⟩ ⊂ C[a, b]. (2.42)

We call the ideal Isym the symmetry ideal or the Sibirsky ideal for family (2.34).The following statement is proved in [56].

Proposition 2.3.2. The focus quantities of systems (2.14) and (2.15) have the form

gkk =∑

ν:L(ν)=(k,k)

g(ν)kk ([ν] − [ν]). (2.43)

By the definition of the Bautin ideal B and this proposition clearly B ⊂ Isym, henceV(Isym) ⊂ V(B), which means every time-reversible system (2.14) or (2.15) has a center atthe origin, so Theorem 2.3.1 holds.

The next theorem gives the exact characterization of the set of time-reversible systems. Aproof can be found in [57].

Theorem 2.3.3 ([53]). Let R ⊂ E(a, b) be the set of all time-reversible systems in family (2.14).Then

(i) R ⊂ V(Isym) and

(ii) V(Isym)\R = (a, b) ∈ V(Isym) : there exists (p, q) ∈ S with apqbqp = 0 but apq +bqp = 0.

From (2.38) and Theorem 2.3.3(i) we have that V(I) ⊂ V(Isym). In fact, it is proven in [53]that

Isym = I and both ideals are prime. (2.44)

From (2.38) and (2.44) we obtain the following statement describing time-reversible systems.

Theorem 2.3.4. The variety of the Sibirsky ideal Isym is the Zariski closure of the set R of alltime-reversible systems in the family (2.34).

The algorithm below provides a generating set for the ideal I (= Isym) and the Hilbert basisfor the monoid M corresponding to system (2.34).

1. Compute a Grobner basis GH for H defined by (2.37) with respect to any eliminationorder with w, γ, tk > apkqk , bqkpk | k = 1, . . . , ℓ;

2. the set GH ∩ k[a, b] is a generating set for I;

3. MH = ν, ν : [ν] − [ν] ∈ G ∪ ej + e2ℓ−j+1 : j = 1, . . . , ℓ and ± ([ej ] − [e2ℓ−j+1]) /∈ Gis a Hilbert basis for the monoid M (here ej is the element (0, . . . , 0, 1, 0, . . . , 0), with thenon-zero entry in the j-th position).

The correctness of the algorithm is proved in [56].


2.3.2 Hamiltonian systems and Darboux integrability

Now we present two other mechanisms for proving the existence of first integrals for polynomialsystems of differential equations on C2 (or R2). We thus consider systems

x = P (x, y), y = Q(x, y), (2.45)

where x, y ∈ C, P and Q are polynomials without constant terms that have no nonconstantcommon factor, and m = max(deg(P ), deg(Q)).

A system (2.45) is said to be a Hamiltonian system if there is a function H : C2 → C, calledthe Hamiltonian of the system, such that P = −Hy and Q = Hx. It follows immediately that

∂H

∂xP +

∂H

∂yQ ≡ 0,

so that H is a first integral of the system (2.45). It is clear that every Hamiltonian system ofthe form (2.23) has a center at the origin.

By the definition a Darboux factor of system (2.45) is a polynomial f(x, y) such that

∂f

∂xP +

∂f

∂yQ = Kf,

where K(x, y) is a polynomial of degree at most n − 1 (Ki(x, y) is called the cofactor). Asimple computation shows that if there are Darboux factors f1, f2, . . . , fk with the cofactorsK1,K2, . . . ,Kk satisfying

k∑i=1

αiKi = 0, (2.46)

then H = fα11 · · · fαk

k , is a first integral of (2.45), and if

k∑i=1

αiKi + P ′x + Q′

y = 0 (2.47)

then the equation admits the integrating factor

µ = fα11 · · · fαk

k . (2.48)

The definition of Darboux integrating factor is consistent with the classical definition of anintegrating factor, which means that once we have Darboux integrating factor, we can integratethe system and find a first integral. The trajectories of system (2.45) satisfy the equation

Q(x, y)dx− P (x, y)dy = 0,

so that µ is an integrating factor in the classical sense if and only if

∂(µQ)

∂y− ∂(−µP )

∂x≡ 0.


Using the integrating factor µ in (2.48) we can compute a first integral by integrating µQ withrespect to x and obtain

H(x, y) = F (x, y) +G(y),

where G is a constant of integration; since we integrate by x it can depend on y. To obtainfunction G(y) we solve the equation

∂H(x, y)

∂y=∂(−µP )

∂y.

Sometimes we can apply the following statement regarding integrating factors. A function V isan inverse integrating factor if V −1 is an integrating factor.

Lemma 2.3.5. (i) If system (2.15) has a local inverse integrating factor

V = (xy)αm∏i=1

F βii

with Fi analytic in x and y, Fi(0, 0) = 0 for i = 1, . . . ,m, α = 0, and α not an integer greaterthan 1, then it has a first integral of the form (2.24).

(ii) If system (2.15) has a local inverse integrating factor V = (xy)α and aα−1,α−1 =bα−1,α−1 = 0 then it has a first integral of the form (2.24).

Part (i) of the lemma is a particular case of Theorem 4.13 (iii) of [21] and part (ii) followsfrom formula (4.28) of [21].

In order to find the Darboux factors of system (2.45) we look for them in the form F =∑si+j=0 αijx

iyj with K =∑m−1

i+j=0 βijxiyj (m is the degree of the system). Substituting these

expressions into∂F

∂xx+

∂F

∂yy = KF, (2.49)

and equating the coefficients of the same terms on both sides of (2.49) we obtain a system ofpolynomial equations in the variables αij , βij . If a solution of the system exists, then solving thesystem we find a Darboux factor F (or few such factors) of (2.45). Geometrically, the equationF = 0 defines a trajectory of (2.45) defined by F (x, y) = 0. Since F is a polynomial, thetrajectory is called an algebraic invariant curve of (2.45).


2.4 Local integrability of a subfamily of the cubic system

In this and following sections we apply the methods and algorithms described above to study theproblem of local integrability for a few families of polynomial autonomous systems of ODE’s.

We consider the complex system

x = x(1 − a10x− a20x2 − a11xy − a02y

2),

y = −y(1 − b01y − b02y2 − b11xy − b20x

2).(2.50)

The following theorem gives the conditions for the origin of family (2.50) to be a center.

Theorem 2.4.1. Let V(B) be the variety of the Bautin ideal of family (2.50) and let B4 =⟨g11, g22, g33, g44⟩. Then V(B) = V(B4). Moreover, V(B) consists of the following five irre-ducible components

V(B) = V(J1) ∪V(J2) ∪V(J3) ∪V(J4) ∪V(J5),

where

J1 = ⟨a210b02 − a20b201, a02a20 − b20b02, a02a

210 − b20b

201, a11 − b11⟩

J2 = ⟨a20 + b20, a11 − b11, a02 + b02⟩J3 = ⟨a11, a02, b02, b11⟩J4 = ⟨a20, a11, b11, b20⟩J5 = ⟨a11, a02, b11, b20⟩.

Proof. Necessity. Using the approach described in Section 2.2 we computed the first sevenfocus quantities g11, . . . , g77 of system (2.50). We have verified the computations also using amodification of Mathematica code given in Appendix B. The first four focus quantities for thesystem (2.50) are as follows:

g11 = b11 − a11,

g22 = − a02a20 + 6a10a11b01 − 6a10b01b11 + b02b20,

g33 = − 15a02a210a11 − 7a02a11a20 + 24a10a

211b01 + 20a02a10a20b01 − 108a210a11b

201 − 4a11a20b

201

− 3a210a11b02 + 4a11a20b02 + 16a02a210b11 + 5a02a20b11 + 108a210b

201b11 + 3a20b

201b11

+ 4a210b02b11 − 4a20b02b11 − 24a10b01b211 + 8a02a11b20 − 16a11b

201b20 − 5a11b02b20

− 20a10b01b02b20 − 8a02b11b20 + 15b201b11b20 + 7b02b11b20,

g44 = − 1944a02a210a

211 − 150a202a

210a20 − 243a02a

211a20 − 162a202a

220 + 7128a02a

310a11b01

+ 4320a10a311b01 + 3832a02a10a11a20b01 − 10800a210a

211b

201 − 4752a02a

210a20b

201

(2.51)

2.4 Local integrability of a subfamily of the cubic system 35

−1584a211a20b201 − 270a02a

220b

201 + 26640a310a11b

301 + 1668a10a11a20b

301

+1296a210a211b02 − 258a02a

210a20b02 + 135a211a20b02 + 108a02a

220b02

+1368a310a11b01b02 − 1412a10a11a20b01b02 + 2574a02a210a11b11 + 90a02a11a20b11

−7428a02a310b01b11 − 10800a10a

211b01b11 − 2068a02a10a20b01b11 + 2826a11a20b

201b11

−26640a310b301b11 − 1368a10a20b

301b11 − 2826a210a11b02b11 − 1668a310b01b02b11

+1412a10a20b01b02b11 − 576a02a210b

211 + 117a02a20b

211 + 10800a10a11b01b

211

+10800a210b201b

211 − 1296a20b

201b

211 + 1584a210b02b

211 − 135a20b02b

211

−4320a10b01b311 + 12a202a

210b20 + 225a02a

211b20 + 180a202a20b20

−1328a02a10a11b01b20 + 576a211b201b20 − 174a02a20b

201b20 + 7428a10a11b

301b20

+174a02a210b02b20 − 117a211b02b20 + 2068a10a11b01b02b20 + 4752a210b

201b02b20

+258a20b201b02b20 + 270a210b

202b20 − 108a20b

202b20 + 1328a02a10b01b11b20

−2574a11b201b11b20 − 7128a10b

301b11b20 − 90a11b02b11b20 − 3832a10b01b02b11b20

−225a02b211b20 + 1944b201b

211b20 + 243b02b

211b20 − 12a02b

201b

220

−180a02b02b220 + 150b201b02b

220 + 162b202b

220.

Reducing g55, g66 and g77 modulo a Grobner basis of the ideal B4 = ⟨g11, g22, g33, g44⟩ we obtain

g55, g66, g77 ≡ 0 mod B4.

Using the Radical Membership Test one can verify that

g44 /∈√

⟨g11, g22, g33⟩.

This means that√

B1 $√

B2 $√

B3 $√

B4 =√B5 = · · · and leads us to expect that

V(B4) = V(B). The inclusion V(B) ⊂ V(B4) is obvious. To verify the opposite inclusion wefind the irreducible decomposition of V(B4) and then check that any point of each componentcorresponds to a system having a center at the origin. To find the irreducible decomposition ofV(B4) we performed computations with the procedure minAssGTZ from the library primdec.lib

[25] of Singular [34] using the degree-reverse lexicographic order with a02 > a10 > a11 > a20 >b02 > b11 > b01 > b20. The code for carrying out the decomposition is as follows.

> LIB "primdec.lib";

> ring r=0,(a02,a10,a11,a20,b02,b11,b01,b20),dp;

> poly g1=a11-b11;

> poly g2=a20*a02-b20*b02;

> poly g3=-1/2*a10^2*a02*b11-...;

> poly g4=-1/3*a10^2*a02^2*b20-2/3*a10^2*a02*b20*b02-...;

> ideal i=g1,g2,g3,g4;

> list L=minAssGTZ(i);

> int k;

> for(k=1;k<=size(L);k++)

> L[k]=std(L[k]);;

> L;


The first command downloads the Singular library for computing the primary and prime de-compositions. The second command declares that the polynomial ring has characteristic zero,that the variables are ordered by a02 > a10 > a11 > a20 > b02 > b11 > b01 > b20 and that theterm order in use is the degree-reverse lexicographic order (dp). In the next four lines we definepolynomials in question and then in the next line ideal is the declaration that we investigatean ideal B4 = ⟨g1, g2, g3, g4⟩. Finally, the command minAssGTZ computes the irreducible de-composition of V(B4) (minimal associate primes of B4). In the last four lines we compute theGrobner basis of each output ideal. We obtain the list of ideals

[1]:

_[1]=a11-b11

_[2]=a20*a02-b20*b02

_[3]=a10^2*b02-a20*b01^2

_[4]=a10^2*a02-b20*b01^2

[2]:

_[1]=a02+b02

_[2]=a11-b11

_[3]=a20+b20

[3]:

_[1]=b02

_[2]=b11

_[3]=a02

_[4]=a11

[4]:

_[1]=b11

_[2]=b20

_[3]=a02

_[4]=a11

[5]:

_[1]=b11

_[2]=b20

_[3]=a11

_[4]=a20

which are exactly the ideals Ji, i = 1, 2, 3, 4, 5 from the statement of Theorem 2.4.1.

Sufficiency. Now we verify that each point of V(Jk), k = 1, 2, 3, 4, 5 corresponds to a systemwith a center at the origin.

Component V(J1). We will show that J1 = Isym. Using the algorithm described in Section2.3 for computing the generators of the ideal I = Isym we have to compute a Grobner basis Gfor the ideal H defined by (2.37), which in our case is

H = ⟨1 − wγ2, a02 − t1, a10 − t2, a11 − t3, a20 − t4, γ2b20 − t1, b01 − γt2, b11 − t3, b02 − γ2t4⟩.

Using the Singular code

> LIB "elim.lib";


> ring r=0,(w,y,t1,t2,t3,t4,a02,a10,a11,a20,b02,b11,b01,b20),dp;

> ideal H=1-w*y^2,a02-t1,a10-t2,a11-t3,a20-t4,y^2*b20-t1,b01-y*t2,b11-t3,

b02-y^2*t4;

> ideal GH=eliminate(H,w*y*t1*t2*t3*t4);

> GH;

we compute in the fourth line a Grobner basis GH of the ideal H ∩C[a, b] containing generatorsfor I = Isym. The command eliminate eliminates polynomials depending on w, γ, t1, t2, t3, t4, itmeans, in our case it computes the 6th-elimination ideal for the order of variables w, γ, t1, t2, t3,t4 ≻ a02, a10, a11, a20, b02, b11, b01, b20 and gives as the output the polynomials that do notdepend on w, γ, t1, t2, t3, t4. The first command LIB "elim.lib" downloads the Singularlibrary for the procedures of elimination, saturation and blowing up. The output for GH is thelist of four polynomials

GH[1]=a11-b11

GH[2]=a02*a20-b02*b20

GH[3]=a10^2*b02-a20*b01^2

GH[4]=a02*a10^2-b01^2*b20.

Thus, the Zariski closure of all time-reversible systems in the family (2.50) is the variety of theideal

Isym = ⟨a11 − b11, a02a20 − b02b20, a210b02 − a20b

201, a

210a02 − b20b

201⟩.

Now we see that V(J1) is the variety of the Sibirsky ideal of (2.50). Thus, by Theorem 2.3.1every system from V(J1) has a center.

Component V(J2). System (2.50) that corresponds to component V(J2) can be written as

x = x(1 − a10x+ b20x2 − b11xy + b02y

2),

y = −y(1 − b20x2 − b01y − b11xy − b02y

2).(2.52)

This system has two invariant algebraic curves l1 = x and l2 = y yielding the integrating factorµ = x−2y−2 and the first integral

H =1

xy(1 − a10x− b01y + b20x

2 − b02y2) + b11 log(xy).

Thus H is of the form

H =R

xy+ c log(xy),

where c = b11 and R = 1 + O(x, y). We claim that there exists an analytic function S =1 + O(x, y) such that

R

xy+ c log(xy) =

S

xy+ c log

(xyS

).

To prove the claim we consider

K(x, y, S) := (R− S) + cxy log(S) = 0. (2.53)

We note that K(0, 0, 1) = 0 and dKdS (0, 0, 1) = 0. Therefore, by the implicit function theorem

there is a function S = 1 + O(x, y), which gives the solution to (2.53) in a neighborhood of the


origin. This proves the claim. Now it is easy to see, by differentiating the display above (2.53)with respect to t and factoring that H = xy/S is an analytic first integral of system (2.52)1.

Component V(J3). We shall show that the system

x = x(1 − a10x− a20x2), y = −y(1 − b20x

2 − b01y) (2.54)

has a center at the origin, i.e. it admits a first integral of the form (2.24). Equivalently, we showthat it admits a first integral of the form

Ψ =

∞∑k=1

fk(y)xk, (2.55)

where, for k ∈ N, fk(y) is an analytic function in the neighborhood of 0. Let fk−1 =b20yf′k−2(y)−

(k − 1)a10fk−1(y). It is easy to see that Ψ = 0 if and only if

f1(y) + y(b01y−1)f ′1(y) = 0 (2.56)

−a10f1(y)+2f2(y) + y(b01y−1)f ′2(y) = 0 (2.57)

−(k − 2)a20fk−2(y)+fk−1(y)+kfk(y)+y(b01y−1)f ′k(y) = 0 (2.58)

for k ≥ 3.We claim, and show by induction that fk(y) in (2.56)–(2.58) are of the form

fk =Pk(y)

(b01y − 1)k, for k = 1, 2, . . . , (2.59)

where Pk(k) is a polynomial of C[y] of degree at most k. Integrating (2.56) and (2.57) we get

f1(y) =yc1

1 − b01y, and f2(y) =

y(a10 + yc2)

(1 − b01y)2, c1, c2 ∈ C,

which proves our claim for k = 1, 2. Setting c1 = 1, c2 = 0 and assuming our claim is true fork = 1, 2, . . . , j−1, we show that it is also true for k = j. Consider (2.58) for k = j and note thatfj−1(y) is of the form (2.59) with k = j − 1. Therefore for k = j, equation (2.58) is equivalentto

y(b01y − 1)f ′j + kfj =Qj−1(y)

(b01y − 1)j−1,

where Qj−1(y) ∈ C[y] of degree at most j − 1. Equivalently

y(b01y − 1)jf ′j + j(b01y − 1)j−1fj = Qj−1(y). (2.60)

To conclude, we need to show that fj(y) in (2.60) has the form (2.59). If

fj(y) =Pj(y)

(b01y − 1)j, then f ′j(y) =

P ′j(y)(b01y − 1) − jb01Pj(y)

(b01y − 1)j+1.

Substituting the above expressions to (2.60), we get

yP ′j(y) − jPj(y) = Qj−1(y).

1We thank prof. C. Christopher for suggesting this proof.


Note that this equation can be easily solved for Pj . Computing we find that the function fj(y)is of the form (2.59), which proves the claim. It can happen that integral (2.55) is only a formalfirst integral of (2.54), however by (d) of Theorem 2.2.2 it is sufficient for the existence of acenter at the origin of (2.54).

Component V(J4). This case is similar to the previous one under the involution akj ↔ bkj .

Component V(J5). Now we consider the system

x = x(1 − a10x− a20x2), y = −y(1 − b01y − b02y

2). (2.61)

It has the following invariant curves

l1 = x,

l2 = y,

l3 = 1 − b01y − b02y2,

l4 = 1 − a10x− a20x2,

with the corresponding cofactors

k1 = 1 − a10x− a20x2,

k2 = −1 + b01y + b02y2

k3 = −y(−b01 − 2b02y),

k4 = x(−a10 − 2a20x).

We can construct an inverse integrating factor of the form V = xyl3l4. Therefore, by Lemma2.3.5 system (2.61) admits a first integral Ψ of the form (2.24).


2.5 Local integrability of a quartic Lotka–Volterra system

In this subsection we obtain the necessary and sufficient conditions for a complex center at theorigin (that is, conditions for the system to have a local first integral of the form (2.24)) for thesystem

x = x(1 − a30x3 − a21x

2y − a12xy2 − a03y

3),y = −y(1 − b30x

3 − b21x2y − b12xy

2 − b03y3).

(2.62)

This eight-parameter quartic system contains some families studied in [14, 17]. To the best ofour knowledges it is the largest subfamily of the quartic system that has been studied so far jointwith the family studied in [17]. System (2.62) has two complex invariant lines passing throughthe origin, which is why we call it a Lotka-Volterra complex system.

To solve the problem we have computed the first focus quantities gjj up to j = 8. Thesefocus quantities are too long, so we do not present them here. However, one can easily checkour calculations with help of any available computer algebra system. A Mathematica code tocompute the focus quantities is presented in Appendix C. Note that if for system (2.62) we havea12 = 0 and b21 = 0, then by a linear transformation

X = ax, Y = by,

where a = a1/312 b

2/321 and b = a

2/312 b

1/321 , we can set in (2.62) a12 = b21 = 1. Using this observation

in order to simplify calculations we split system (2.62) into three systems considering separatelythe cases:

(α) a12 = b21 = 1, (β) a12 = 1, b21 = 0, (γ) a12 = b21 = 0.

For the case (α) we have obtained necessary conditions to have a complex center at the originpresented in Theorem 2.5.1. We have also proved the sufficiency of these conditions except forcondition 10). The other remaining cases are studied in Theorems 2.5.2 and 2.5.3. Notice thatif we apply to the conditions of Theorem 2.5.2, which give the conditions for the case (β), theinvolution aij ↔ bji, then we obtain the integrability conditions of system (2.62) for the casea12 = 0, b21 = 1. Thus, Theorems 2.5.1–2.5.3 provide the solution to the complex center problemof system (2.62).

Theorem 2.5.1. The following conditions are necessary conditions for system (2.62) with a12 =b21 = 1 to have a complex center at the origin:

1) a21−b12 = 2a03b12+b12b03−a03−2b03 = a30b12+2b30b12−2a30−b30 = a30a03−b30b03 = 0;

2) a30 − b30 = a21 − b12 = a30 − b03 = 0;

3) b03 = b12 − 2 = a03 = a21 − 2 = 0;

4) b12 − 2 = b30 = a21 − 2 = a30 = 0;

5) b12 − 2 = a03 − 2b03 = a21 − 2 = 2a30 − b30 = 0;

6) b03 − 1 = b30 + b12 − 2 = a03 − b12 + 2 = a21 + b12 − 4 = a30 − 1 = 0;

7) 12a03 − 8b12 − 3b03 + 7 = 3a30 + 8a21 − 12b30 − 7 = 2b30b03 − a21 + b12 = 0,8a21b03 + 12b12 − 7b03 + 3 = 32b212 − 20b12 + 27b03 − 7 = 8b30b12 + 3a21 − 7b30 − 6 = 0,8a21b12 − 7a21 − 7b12 − 4 = 8a221 − 23a21 + 27b30 + 14 = 0;

2.5 Local integrability of a quartic Lotka–Volterra system 41

8) b03 − 4 = b12 + 9 = 4b30 − 5 = a03 − 8 = a21 + 2 = 4a30 + 1 = 0;

9) 4b03 + 1 = b12 + 2 = b30 − 8 = 4a03 − 5 = a21 + 9 = a30 − 4 = 0;

10) 9b03 + 16 = 3b12 + 5 = 16b30 + 9 = 9a03 + 8 = 2a21 + 1 = 16a30 − 3 = 0;

11) 4b03 − 1 = b12 − 1 = b30 + 4 = a03 = a21 = a30 + 8 = 0;

12) 16b03 − 3 = 2b12 + 1 = 9b30 + 8 = 16a03 + 9 = 3a21 + 5 = 9a30 + 16 = 0;

13) b03 + 8 = b12 = b30 = a03 + 4 = a21 − 1 = 4a30 − 1 = 0.

If one of conditions 1)-9), 11)–13) holds, then the corresponding system has a center at theorigin.

Proof. To obtain the necessary conditions for the existence of a center of system (2.62) witha12 = b21 = 1 we look for the irreducible decomposition of the variety of the ideal I8 =⟨g11, g22, . . . , g88⟩, where a12 = b21 = 1. This is an extremely difficult computational problemwhich represents the most laborious part of our study.

Making use of the routine minAssGTZ of computer algebra system Singular, we wereunable to find the irreducible decomposition of the variety of the ideal I8 = ⟨g11, g22, . . . , g88⟩over the field of rational numbers, but we have found it using the Decomposition Algorithmwith Modular Arithmetics described in Subsection 1.2.3. Computing in the field of characteristic32003 we obtained the decomposition consisting of 13 components defined by the following primeideals:

1) ⟨a21−b12, a03b12−16001b12b03+16001a03−b03, a30b12+2b30b12−2a30−b30, a30a03−b30b03⟩,

2) ⟨a03 − b30, a21 − b12, a30 − b03⟩,

3) ⟨b03, b12 − 2, a03, a21 − 2⟩,

4) ⟨b12 − 2, b30, a21 − 2, a30⟩,

5) ⟨b12 − 2, a03 − 2b03, a21 − 2, a30 + 16001b30⟩,

6) ⟨b03 − 1, b30 + b12 − 2, a03 − b12 + 2, a21 + b12 − 4, a30 − 1⟩,

7) ⟨a03 + 10667b12 − 8001b03 − 13334, a30 − 10665a21 − 4b30 − 10670, b30b03 + 16001a21 −16001b12, a21b03 − 16000b12 − 12002b03 − 4000, b212 − 4001b12 − 9000b03 + 13001, b30b12 −4000a21−12002b30+8000, a21b12−12002a21−12002b12+16001, a221−12004a21−3997b30−7999⟩,

8) ⟨b03 − 4, b12 + 9, b30 − 8002, a03 − 8, a21 + 2, a30 + 8001⟩,

9) ⟨b03 + 8001, b12 + 2, b30 − 8, a03 − 8002, a21 + 9, a30 − 4⟩,

10) ⟨b03 − 7110, b12 − 10666, b30 − 6000, a03 − 3555, a21 − 16001, a30 + 2000⟩

11) ⟨b03 − 8001, b12 − 1, b30 + 4, a03, a21, a30 + 8⟩

12) ⟨b03 + 2000, b12 − 16001, b30 − 3555, a03 − 6000, a21 − 10666, a30 − 7110⟩,


13) ⟨b03 + 8, b12, b30, a03 + 4, a21 − 1, a30 − 8001⟩.

Now we use the rational reconstruction algorithm described in Section 1.1 for the 13 ideals givenabove and obtain the ideals in Theorem 2.5.1 which we denote by Qi, i = 1, . . . , 13. A simpleMathematica code to perform the rational reconstruction following the algorithm described inSubsection 1.2.3 is given in Appendix A. Following the Decomposition Algorithm with ModularArithmetics we first check that gkk are in radicals of the ideals Qi, that is, the reduced Grobnerbasis of each ideal ⟨1−wgkk, Qi⟩, where k = 1, . . . , 8 and i = 1, . . . , 13, is 1. Since the answeron the third step of the Decomposition Algorithm with Modular Arithmetics is ”yes” we can goto the fourth step of the algorithm to check that no condition is lost under the computations. Wecompute Q = ∩13

i=1Qi. With the routine intersect of Singular working in the field of rationalnumbers we easily compute the ideal Q and find that the output list contains 28 polynomials,which we denote q1, . . . , q28. Then, to check that

√Q =

√I8, we compute a Grobner basis of each

ideal ⟨1−wqi, I8⟩ for i = 1, . . . , 28 and a Grobner basis of each ideal ⟨1−wgkk, Q⟩, k = 1, . . . , kand it turns out, that all Grobner bases are 1. This means that V(I8) = ∪13

i=1V(Qi).We now show that under each of these conditions the system has a complex center.

For Case 1) the corresponding system is written as

x = x(1 − a30x3 − b12x

2y − xy2 − a03y3),

y = −y(

1 − a30(2−b12)2b12−1 x3 − x2y − b12xy

2 − a03(2b12−1)2−b12

y3).

(2.63)

Using the Darboux method described in Subsection 2.3.2 we have found that the system hasthe following inverse integrating factor

V = (xy)− 2−3b12

b12−1 ,

which has the associated first integral

H = (xy)1−2b12b12−1 (2 − b12 + a30(b12 − 2)x3 + (2 − 5b12 + 2b212)(x

2y + xy2) + a03(1 − 2b12)y3).

We see that H is not defined for b12 = 1 and the expression on the right-hand side of thesecond equation in system (2.63) is not defined for b12 = 2 and b12 = 1/2. Let us denote by Vthe variety V(I), where I is the ideal generated by polynomials from conditions of case 1) ofTheorem 2.5.1,

I = ⟨a21 − b12, 2a03b12 + b12b03 − a03 − 2b03, a30b12 + 2b30b12 − 2a30 − b30, a30a03 − b30b03⟩,

and by J1 = ⟨b12 − 1⟩, J2 = ⟨b12 − 2⟩ and J3 = ⟨b12 − 1/2⟩. We see that H is defined for allpoints from V(I) except maybe for points where b12 is 1, 2 or 1/2, i.e. it is defined for pointsfrom V \V(J) where J = J1 ∩ J2 ∩ J3. Since V \V(J) is not necessarily a variety, the smallestvariety that contains it is its Zariski closure V \V(J). From the Theorem 1.2.14 we know that

V(I : J) = V(I)\V(J).

Using the Singular routine intersect we first compute the ideal J and obtain J = ⟨2b312 −7b212 + 7b12 − 2⟩, and then using the routine quotient we compute the ideal quotient of I andJ and we obtain the ideal

I : J = ⟨a21 − b12, 2a03b12 + b12b03 − a03 − 2b03, a30b12 + 2b30b12 − 2a30 − b30, a30a03 − b30b03⟩.


We see that I : J = I, therefore,

V \V(J) = V

and thus, for each point of V , the corresponding system (2.62) has a center at the origin2.

Since in Case 2) the conditions of time-reversibility given in [17] are fulfilled systems of thiscase are time-reversible systems.

The system of Case 3) is case 6) of Theorem 2 in [31].


In Case 5) the system (2.62) has the form

x = x(

1 − b302 x

3 − 2x2y − xy2 − 2b03y3),

y = −y(1 − b30x3 − x2y − 2xy2 − b03y

3).

and admits the invariant curve

f = (4 − 24x2y − 24xy2 + 36x4y2 + 18b30x4y2 + 108x3y3 − 54b03b30x

3y3

−54b30x6y3 + 27b03b

230x

6y3 + 36x2y4 + 36b03x2y4 − 108x5y4 + 54b03b30x

5y4

−108x4y5 + 54b03b30x4y5 − 108b03x

3y6 + 54b203b30x3y6),

with the corresponding cofactor k = −6x(x− y)y. An inverse integrating factor is given by

V = (xy)−2f.

The existence of an analytic first integral in a neighborhood of the origin follows from statement(i) of Lemma 2.3.5.


Case 7) was solved in [31].

In Case 8) the system (2.62) has the form

x = x(1 + x3/4 + 2x2y − xy2 − 8y3),

y = −y(1 − 5x3/4 − x2y + 9xy2 − 4y3),

and admits the invariant curves

f1 = 1 + x3 + 3x6/8 + x9/16 + x12/256 + 8x2y + 3x5y − x11y/16 + 28xy2 + 3x4y2

−15x7y2/4 + 7x10y2/16 − 60x3y3 + 20x6y3 − 7x9y3/4 + 78x2y4 − 45x5y4

+35x8y4/8 + 48x4y5 − 7x7y5 − 20x3y6 + 7x6y6 − 4x5y7 + x4y8,

and

f2 = 1 + 3x3/4 + 3x6/16 + x9/64 + 6x2y + 3x5y/4 − 3x8y/16 + 21xy2 − 9x4y2

+15x7y2/16 + 21x3y3 − 5x6y3/2 − 15x2y4 + 15x5y4/4 − 3x4y5 + x3y6,

2With similar computations we can prove integrability for systems, which we study below in cases when theDarboux integral or integrating factor is not defined.


with the corresponding cofactors k1 = x(x − 2y)(3x + 14y) and k2 = 3x(x − 2y)(3x + 14y)/4,respectively. A Darboux first integral is given by

H(x, y) = f−31 f42 .

Case 9) is the dual case of the case 8) under the involution akj ↔ bjk.

Case 11) In this case system (2.62) has the form

x = x(1 + 8x3 − xy2),y = −y(1 + 4x3 − x2y − xy2 − y3/4),

(2.64)

and admits the invariant curves

f1 = 1 − 3xy2/2 − y3/4, and f2 = 1 + 8x3 − 3xy2/2 − y3/4,

with the corresponding cofactors k1 = 3y2(2x+ y)/4 and k2 = 3(4x+ y)(8x2 − 2xy+ y2)/4. Aninverse integrating factor is given by

V (x, y) = x2y2f5/61 f

−1/62 .

To prove the existence of an analytic first integral in a neighborhood of the origin we use

statement (ii) of Lemma 2.3.5. Multiplying both parts of (2.64) by f5/61 f

−1/62 we obtain a

systemx = P (x, y), y = Q(x, y), (2.65)

such that V = x2y2 is an inverse integrating factor of (2.65) and the coefficients of xy in theseries expansions for P and Q, respectively, are equal to zero. Therefore, by Lemma 2.3.5 thesystem admits a first integral Ψ of the form (2.24).



Remark. We were not able to prove the integrability of the system of case 10).To obtain the necessary conditions for cases (β) and (γ), presented in Theorem 2.5.2 and

2.5.3, respectively, we found the minimal associate primes of the corresponding ideals I8 =⟨g11, g22, . . . , g88⟩. For these ideals we were able to complete all the calculations with Singularover the field of rational numbers, so the modular calculations were not involved.

Theorem 2.5.2. System (2.62) with a12 = 1, b21 = 0 has a complex center at the origin if andonly if one of the following conditions holds:

1) a21 = 2a03b12 + b12b03 − a03 − 2b03 = a30b12 + 2b30b12 − 2a30 − b30 = a30a03 − b30b03 = 0;

2) b03 = b12 − 2 = a03 = a21 = 0;

3) b30 = a21 = a30 = 0;

4) b12 − 2 = a03 − 2b03 = a21 = 2a30 − b30 = 0;

5) 8b12 − 7 = 4a03 + b03 = a30 − 4b30 = 2b30b03 − a21 = 16a21b03 + 27 = 8a221 + 27b30 = 0;

6) b12 = b30 = 2a03 + b03 = a30b03 − 2a21 = a21b03 + 8 = a221 + 4a30 = 0;


7) b12 = b30 = 2a03 − b03 = a30b03 + 2a21 = a21b03 − 8 = a221 + 4a30 = 0.

Proof. We now prove that under each of conditions 1)-7) the corresponding systems have centersat the origin.

In Case 1) system (2.62) takes the form

x = x(

1 − a30x3 − xy2 + b03(b12−2)

2b12−1 y3),

y = −y(

1 + a30(b12−2)2b12−1 x3 − b12xy

2 − b03y3).

(2.66)

System (2.66) has the cubic invariant curve f1 = 1 − a30x3 + (1 − 2b12)xy

2 − b03y3, with the

corresponding cofactor k1 = −3a30x3−xy2 +2b12xy

2 +3b03y3, and has the analytic first integral

H = xyf1−b122b12−1

1 .



In Case 4) system (2.62) takes the form

x = x(1 − b30x3/2 − xy2 − 2b03y

3), y = −y(1 − b30x3 − 2xy2 − b03y

3). (2.67)

System (2.67) has the following invariant curve

f1 = 4−24xy2+18b30x4y2−54b03b30x

3y3+27b03b230x

6y3+36x2y4+54b03b30x4y5+54b203b30x

3y6,

with the corresponding cofactor k1 = 6xy2. An inverse integrating factor for system (2.67) isgiven by V = x−2y−2f1. The existence of an analytic first integral in a neighborhood of theorigin is guaranteed by statement (i) of Lemma 2.3.5.

In Case 5), system (2.62) takes the form

dx

dt= x+

27x4

128a203− 27x3y

64a03− x2y2 − a03xy

3,

dy

dt= −y − 27x3y

512a203+

7xy3

8− 4a03y

4.

(2.68)

System (2.68) has the cubic invariant algebraic curve

f1 = 1 +27x3

128a203− 9x2y

16a03− 3xy2

2+ 4a03y

3,

with the cofactor

k1 =3(3x− 8a03y)(9x2 + 16a03xy + 64a203y

2)

128a203.

However in this case we are not able to find more invariant curves other than the coordinateaxes and using these curves we are not able to find a Darboux integral or integrating factor. We


use another method to prove the sufficiency of the condition. We introduce new variables z andw by setting

z3 =512a303y

3

3(1024a203 + 135x3 − 216a03x2y − 960a203xy2 + 1536a303y

3),

w3 =−9x3

1024a203 + 135x3 − 216a03x2y − 960a203xy2 + 1536a303y

3).

(2.69)

Then,

x3 =−1024a203w

3

9(1 − 9z3 − 15z2w + 9zw2 + 15w3),

y3 =6z3

a03(1 − 9z3 − 15z2w + 9zw2 + 15w3).

(2.70)

After the change of time

dt = (1 − 9z3 − 15z2w + 9zw2 + 15w3)dT

we obtain the systemdz

dT= −z(1 + h),

dw

dT= w(1 − h), (2.71)

where

h = 3(z − w)(2z2 + 2w2 − 45z5 − 117z4w − 94z3w2 − 94z2w3 − 117zw4 − 45w5).

Obviously, system (2.71) is unchanged under the transformation

z → w, w → z, T → −T,

that is, it is a so-called time-reversible system. Hence, by Theorem 3.5.5 of [56] system (2.71)has a first integral Ψ = zw + . . . 3.



Remark. Some functions appearing in the proofs of Theorem 2.5.1 and Theorem 2.5.2 arenot defined for specific values of parameters. The existence of analytic first integrals for thesespecific values follows from the fact that the set of all systems (2.62) with a complex center isa closed set in the Zariski topology and similar computations as in the proof of Case 1) of theprevious theorem.

Theorem 2.5.3. System (2.62) with a12 = b21 = 0 has a complex center at the origin if andonly if one of the following conditions holds:

1) b03 = a03 = a21 = 0;

2) a30a03 − b30b03 = a30b312 − a321b03 = a321a03 − b30b

312 = 0;

3The proof for this case was suggested by Y. Liu in [31]


3) b12 = b30 = a30 = 0;

4) 2a03 + b03 = a30 + 2b30 = 0;

5) a03 − 2b03 = 2a30 − b30 = 2a21b12 − 9b30b03 = 0.

Proof. The system of Case 1) is case 3) of Theorem 4 in [31].

In Case 2), the system is time-reversible.

The Case 3) is the dual to Case 1) under the involution akj ↔ bjk.

In Case 4), system (2.62) has the form

x = x(1 + 2b30x3 − a21x

2y − a03y3), y = −y(1 − b30x

3 − b12xy2 + 2a03y

3).

It has the cubic invariant curve

f1 = 1 + 2b30x3 − 2a21x

2y − 2b12xy2 + 2a03y

3,

with the cofactor k1 = 2(3b30x3 − a21x

2y + b12xy2 − 3a03y

3). Moreover, it has the analytic first

integral H = xyf−1/21 .

In Case 5), system (2.62) becomes

x = x(

1 − a30x3 − a21x

2y − 2a21b129a30 y3

), y = −y

(1 − 2a30x

3 − b12xy2 − a21b12

9a30 y3). (2.72)

System (2.72) has the invariant curves given by

f1 = 1 − 2a21x2y − 2b12xy

2 + 3a30b12x4y2 + 2a21b12x

3y3 +a221b123a30

x2y4,

and

f2 = −18a21a230 + 54a221a

230x

2y + 54a21a230b12xy

2 − 27a321a230x

4y2 − 81a21a330b12x

4y2

−9a421a30x3y3 − 54a221a

230b12x

3y3 − 81a330b212x

3y3 + 27a221a330b12x

6y3 + 81a430b212x

6y3

−9a321a30b12x2y4 − 27a21a

230b

212x

2y4 + 27a321a230b12x

5y4 + 81a21a330b

212x

5y4

+9a421a30b12x4y5 + 27a221a

230b

212x

4y5 + a521b12x3y6 + 3a321a30b

212x

3y6,

with the cofactors k1 = −2xy(a21x − b12y) and k2 = −3xy(a21x − b12y), respectively. Hence aDarboux first integral is given by

H(x, y) = f31 f−22 .


2.6 Local integrability of a quintic system

It is known that the case of homogeneous perturbations of odd degree is easer to tackle thanthe case of homogeneous perturbations of even degree, so it appears worthwhile to study thesystem (2.14) with P and Q being homogeneous quintic polynomials first, that is, to study thecenter problem for the system

x = x− a40x5 − a31x

4y − a22x3y2 − a13x

2y3 − a04xy4 − a−15y

5,

y = −y + b5,−1x5 + b40x

4y + b31x3y2 + b22x

2y3 + b13xy4 + b04y

5,(2.73)

where x, y, aij , bji ∈ C. It turns out the computations involved in the determination of necessaryconditions for integrability for the full family (2.73) are so heavy that they cannot be completedeven using powerful computers and the modern computer algebra systems. Thus, it is reasonableto study some subfamilies of system (2.73). Recently, the center conditions for the subfamily of(2.73), where a−15 = b5,−1 = 0 have been obtained in [32]. By a linear transformation

X = ax, Y = by,

where a = 24

√a−15b55,−1 and b = 24

√a5−15b5,−1, system (2.73) with a−15b5,−1 = 0 can be written

asx = x− a40x

5 − a31x4y − a22x

3y2 − a13x2y3 − a04xy

4 − y5,

y = −y + x5 + b40x4y + b31x

3y2 + b22x2y3 + b13xy

4 + b04y5.

(2.74)

We find necessary conditions for existence of a center at the origin for the following eight-parameter subfamilies of system (2.74):

(C1) a40 = b04 = 0, (C2) a31 = b13 = 0, (C3) a13 = b31 = 0, (C4) a04 = b40 = 0.

For most cases we show that the obtained conditions are also sufficient conditions for existenceof the center.

Computing with the algorithm from Section 2.3 we find that the Zariski closure of all time-reversible systems in the family (2.73) is the variety of the ideal

Isym = ⟨a22 − b22, a40a04 − b40b04, a31a13 − b31b13, a40b213 − a231b04,

a213b40 − a04b231, a

231a04 − b40b

213, a40a

213 − b231b04, a

313b5,−1 − a−1,5b

331,

a331a−1,5 − b5,−1b313, a

304b

25,−1 − a2−1,5b

320, a

340a

2−1,5 − b25,−1b

304⟩.

(2.75)

We give conditions for the existence of the first integral (2.24) for subfamilies (C1)− (C4) ofsystem (2.74), that is, for the existence of a center at the origin for the systems.

2.6 Local integrability of a quintic system 49

Case (C1)

Conditions for the existence of a center for systems of this case are given in the following theorem.

Theorem 2.6.1. For case (C1), system (2.74) is integrable if a22 = b22 and one of the followingconditions holds:(α) a04 − b40 = a13 − b31 = a31 − b13 = 0;(β) b40 = a04 = 2b13 − a13 = 2a31 − b31 = 0;(γ) a204 + a04b40 + b240 = b31a04 + a13b40 + b31b40 = a13a04− b31b40 = a213 + a13b31 + b231 = a31b40−b13a04 = a31a04+b13a04+b13b40 = b13a13+a31b31+b13b31 = a31a13−b13b31 = a231+a31b13+b213 = 0.

Proof. For system (2.73) using the computer algebra system Mathematica we computed 22first focus quantities, g1,1, ..., g22,22, and found that gs,s = 0 for s odd andg2,2 = a22 − b22;g4,4 = a13a31 + a04a40 − b13b31 − b04b40;g6,6 = 192a13a22a31 + 36a213a40 + 57a04a22a40 + 8a15a31a40 − 27a22a40b04 − 144a22a31b13 −12a13a40b13−168a13a31b22−33a04a40b22+27a40b04b22+144a31b13b22−216a13a22b31+36a04a31b31+36a15a40b31 + 12a31b04b31 + 168a22b13b31 + 216a13b22b31 − 192b13b22b31 − 12a04b

231 − 36b04b

231 +

12a213b40 − 63a04a22b40 + 24a15a31b40 + 33a22b04b40 − 36a13b13b40 + 63a04b22b40 − 57b04b22b40 −20a15b31b40 + 20a04a13b51 − 32a15a22b51 − 36a13b04b51 − 24a04b13b51 − 8b04b13b51 + 32a15b22b51.The size of the polynomials gk,k sharply increases so we do not present the other polynomialshere, but the interested reader can easily compute them using any available computer algebrasystem.

Since g2,2 = a22 − b22 = 0 is the necessary condition for integrability of system (2.74) fromnow on we assume that a22 = b22. Then, using the routine minAssGTZ [25] of Singular [34] andperforming computations over the field of characteristic 32003 (we were not able to completecomputations over the field of characteristic 0 on our computational facilities), we found thatthe minimal associate primes of the ideal I18 = ⟨g2,2, ..., g18,18⟩ areJ1 = ⟨a04 − b40, a13 − b31, a31 − b13⟩,J2 = b13 + 16001a13, a31 + 16001b31, b40, a04⟩,J3 = ⟨a204 + a04b40 + b240, b31a04 + a13b40 + b31b40, a13a04 − b31b40, a

213 + a13b31 + b231,−b13a04 +

a31b40, a31a04 + b13a04 + b13b40, b13a13 + a31b31 + b13b31, a31a13 − b13b31, a231 + a31b13 + b213⟩.

After rational reconstruction we obtain from J1, J2, J3 the conditions (α)–(γ) of Theorem 2.6.1.We denote ideals generated by polynomials from conditions (α)–(γ) by Q1, Q2 and Q3. Howeverit is known that modular computations provide a result which might not be correct, but it iscorrect with high probability. If we follow the Decomposition Algorithm with Modular Arith-metics from Section 1.1, we next check that under each condition (α)–(γ) all focus quantitiesgk,k ∈ I18 vanish. Then, to check that no condition is lost, we compute the intersection Q of ideals Q1, Q2 and Q3 in the ring Q[a31, b14, a13, b31, a04, b40, b22] and compute Grobner bases of ideals⟨1−wgkk, Q⟩ and ⟨1−wqi, I18⟩. We were not able to complete computations of the Grobner basesover Q. However the computations over fields of characteristics 32003 and 4236233 shows thatall bases are 1 in the corresponding rings. That means that V(I18) = V(Q) with probabilityclose to one.4

We now prove that under each of conditions (α)–(γ) there is a center at the origin.

4However since still there is a small probability that a component of the variety of V(I18) is lost we say in thestatement of the theorem ”if”, but not ”if and only if”


If condition (α) or (γ) holds, then all polynomials defining the ideal (2.75) vanish. It meansthe corresponding systems are in the variety of the Sibirsky ideal.

When condition (β) hold, system (2.74) is written as

x = x− a31x4y − b22x

3y2 − 2b13x2y3 − y5,

y = −y + x5 + 2a31x3y2 + b22x

2y3 + b13xy4.

It is a Hamiltonian system with the Hamiltonian

H(x, y) = (6xy − x6 − 3a31x4y2 − 2b22x

3y3 − 3b13x2y4 − y6)/6.

Case (C2)

In case (C2), that is when a31 = b13 = 0, substituting in the polynomials of the ideal I18 a22 = b22and computing with minAssGTZ of Singular in the field of characteristic 32003 with the degreereverse lexicographic order with a31 > b13 > a13 > b31 > a04 > b40 > a40 > b04 > b22 we foundthe following components of the variety V(I18):1) a240 +a40b04 + b204 = b40a40 +a04b04 + b40b04 = a04a40− b40b04 = a204 +a04b40 + b240 = −b31a40 +a13b04 = a13a40+b31a40+b31b04 = b31a04+a13b40+b31b40 = a13a04−b31b40 = a213+a13b31+b231 = 0;2)a40 − b04 = a04 − b40 = a13 − b31 = 0;3) b22 = a40 + b04 = b40 = a04 = b31 + 1 = a13 + 1 = 0;4)a240 − a40b04 + b204 = b31b04 − a40 = b31a40 − a40 + b04 = b231 − b31 + 1 = b22 = b40 = a04 =b231 + a13 = 0;5) b40 − 5a40 = a04 − 5b04 = b31 = a13 = 0;6) a40b04 + 5999 = b22 = b40 − 6400a40 = a04 − 6400b04 = b31 = a13 = 0;7) 8686b304b

522+b604−9149b304b

322−11558b622+13631b304b22+4254b422−5589b222+439 = −5120b204b

522−

14532b504+12806b204b322+a40b

422+12878b204b22+6407a40b

222−2550a40 = a40b04−9112b222−3556 =

6934b04b522 + 12830b404 + a240b

222 + 5334b04b

322 − 12798a240 − 10665b04b22 = 8686b522 + a340 + b304 −

9149b322 + 13631b22 = b31b222 + 48a240 − 8b04b22 + 16b31 = 5482b04b

422 + b31b

304 + 8890a240b22 +

3630b04b222 − 8297b31b22 + 11853b04 = −4000b31b

204b22 − 3b304 + 4806b322 + b31a40 + 5778b22 =

12001b31b04b22 + b231 + b204 − 12000a40b22 = b40 − 3a40 = a04 − 3b04 = 4000b31b04b22 + 3b204 −4001a40b22 + a13 = 0.

Performing the rational reconstruction with the algorithm described in Section 1.2.3 weobtain the components defined by the ideals(α) J1 = ⟨a240 + a40b04 + b204, b40a40 + a04b04 + b40b04, a04a40− b40b04, a

204 + a04b40 + b240,−b31a40 +

a13b04, a13a40 + b31a40 + b31b04, b31a04 + a13b40 + b31b40, a13a04 − b31b40, a213 + a13b31 + b231⟩,

(β) J2 = ⟨a40 − b04, a04 − b40, a13 − b31⟩;(γ) J3 = ⟨b22, a40 + b04, b40, a04, b31 + 1, a13 + 1⟩;(δ) J4 = ⟨a240−a40b04 + b204, b31b04−a40, b31a40−a40 + b04, b

231− b31 +1, b22, b40, a04, a13 + b31−1⟩;

(ϵ) J5 = ⟨b40 − 5a40, a04 − 5b04, b31, a13⟩;(ζ) J6 = ⟨a40b04 − 25

16 , b22, b40 + 35a40, a04 + 3

5b04, b31, a13⟩,and the component defined byU = ⟨−37

70b304b

522+b604− 37

7 b304b

322− 49

36b622+ 5

54b304b22+ 69

158b422− 74

63b222+ 44

73 ,1225b

204b

522+ 163

11 b504+ 24

5 b204b

322+

a40b422−103

82 b204b22+

325 a40b

222−123

113a40, a40b04−5

144b222−1

9 ,160b04b

522+

1445 b

404+a

240b

222+

16b04b

322+

165 a

240+


83b04b22,−

3770b

522 + a340 + b304− 37

7 b322 + 5

54b22, b31b222 + 48a240− 8b04b22 + 16b31,−148

35 b04b422 + b31b

304 +

518a

240b22 + 77

97b04b222 + 2

27b31b22 + 127b04,

38b31b

204b22 − 3b304 + 111

20 b322 + b31a40 − 23

72b22,−18b31b04b22 +

b231 + b204 + 98a40b22, b40 − 3a40, a04 − 3b04,−3

8b31b04b22 + 3b204 − 58a40b22 + a13⟩.

Using the radical membership test and computing over the field of characteristic zero wehave checked that V(Ji) ⊂ V(I18) for i = 1, . . . , 6, however V(U) ⊂ V(I18), that is, V(U) isnot a component of a center variety.

A usual procedure in such situations is to look for the decomposition of the variety over afield of another finite characteristic. We repeated the calculations in the fields of characteristics4256233 and 7368787. In both cases the decomposition of the variety consists of 9 components;however after the reconstruction we obtained the components (β), (γ), (ϵ), (ζ). So, the result isworse than the one obtained computing over 32003. From our empirical observation in such caseswhen one component of a modular decomposition is not a true one in the polynomial ring overthe field of characteristic zero, the most simple polynomials defining the associate primes areusually the correct polynomials. So we recomputed with the condition b40−3a40 = a04−3b04 = 0over the field of characteristic zero and found the components:U1 = ⟨3b231+4a40b22−a13, 9a13b31−2b222−1, 6b04b31+a13b22+2a40, 3a

213+4b04b22−b31, 6a40a13+

b31b22 + 2b04, 144a40b04 − 5b222 − 16, b322 − 36b04a13 − 36a40b31 + 14b22, b31b222 + 48a240 − 8b04b22 +

16b31, a13b222 + 48b204 − 8a40b22 + 16a13, b40 − 3a40, a04 − 3b04⟩;

U2 = ⟨a13 − b31 = a40 − b04 = b40 − 3a40 = a04 − 3b04 = 0; ⟩U3 = ⟨a213+a13b31+b231, b04a13−a40b31, a40a13+a40b31+b04b31, a

240+a40b04+b204, b40−3a40, a04−

3b04.⟩Then, with the intersect of Singular we computed J = ∩6

i=1Ji ∩ U1 ∩ U2 ∩ U3 and withminAssChar found that the the minimal associate primes of J are the ideals J1, . . . , J6 and

(η) J7 = ⟨4a313 + 3a213b231 + 6a13b31 + 4b331− 1, 2b222− 9a13b31 + 1, 108b40b

331− 4b40− 12b322a13−

9b322b231−12b22a13−108b22b

231, 4b40a13−12b40b

231+5b322b31+6b22a

213−18b22a13b

231+16b22b31, 4b40b22−

3a13 + 9b231, 16b240 + 24b40b22a13 + 15b222b31 + 48b31, 640a04 − 384b340b31 + 32b240b322 + 2576b240b22 +

4608b40b231 + 75b322b31 + 1200b22b31, 3b04 − a04, 3a40 − b40⟩.

Using the radical membership test we checked that√I18 ⊂

√J (where J = ∩7

i=1Ji ⊂ Q2 =Q[a40, b04, a13, b31, a04, b40, b22]) yielding V(J) ⊂ V(I18). However we failed to complete thecomputation of Grobner bases in Q2 to check that

√J ⊂

√I18. (2.76)

Nevertheless computations show that (2.76) holds in the rings k[a40, b04, a13, b31, a04, b40, b22]with k being the fields of characteristic 32003 and 4256233. This means that with very highprobability (2.76) holds also in Q2.

Theorem 2.6.2. For case (C2), system (2.74) is integrable if a22 = b22 and one of the followingconditions holds:(α) a240+a40b04+b204 = b40a40+a04b04+b40b04 = a04a40−b40b04 = a204+a04b40+b240 = −b31a40+a13b04 = a13a40+b31a40+b31b04 = b31a04+a13b40+b31b40 = a13a04−b31b40 = a213+a13b31+b231 = 0;(β) a40 − b04 = a04 − b40 = a13 − b31 = 0;(γ) b40 − 5a40 = a04 − 5b04 = b31 = a13 = 0.

Proof. When condition (α) or (β) holds the system is in the Zariski closure of the set of time-reversible systems.

When condition (γ) holds, the corresponding system is Hamiltonian and has a first integral

H(x, y) = (6xy − x6 − 6a40x5y − 2b22x

3y3 − 6b04xy5 − y6)/6.


Case (C3)

In a manner similar to that above computing with minAssGTZ the minimal associate primesof the ideal I18 we obtain ten following components:(α) a240 + a40b04 + b204 = b40a40 + a04b04 + b40b04 = a04a40 − b40b04 = a204 + a04b40 + b240 =b13a40 + a31b04 + b13b04 = a31a40 − b13b04 = −b13a04 + a31b40 = a31a04 + b13a04 + b13b40 =a231 + a31b13 + b213 = 0;(β) a40 − b04 = a04 − b40 = a31 − b13 = 0;(γ) b22 + 1 = b13 + 2a40 = −2a40b

222 + b40 + a40 = −2b04b

222 + a04 + b04 = 8a40b

204b22 − 4b13b

204 −

2b04b22 + a31 = 0;(δ) b22 − 1 = b13 − 2a40 = −2a40b

222 + b40 + a40 = −2b04b

222 + a04 + b04 = 8a40b

204b22 − 4b13b

204 −

2b04b22 + a31=0;(ϵ) a40b04 − 1

4 = b13 − 2a40 = b22 = b40 + a40 = a04 + b04 = −4b13b204 + a31 = 0;

(ζ) a40b04 − 14 = b13 + 2a40 = b22 = b40 + a40 = a04 + b04 = −4b13b

204 + a31 = 0;

(η) b40 − 5a40 = a04 − 5b04 = b13 = a31 = 0;(θ) a40b04 − 25

16 = b22 = b40 + 35a40 = a04 + 3

5b04 = b13 = a31 = 0;(ι) b22 + 1 = a40b04 − 1 = b40 = a04 = −a40b22 + b13 = −b04b22 + a31 = 0;(κ)b22 − 1 = a40b04 − 1 = b40 = a04 = −a40b22 + b13 = −b04b22 + a31 = 0.Note that no fake component (like the one defined by the ideal U in case (C2)) appears forthis case (and for case (C4)), so the components 1)–10) are obtained after applying the rationalreconstruction algorithm to the ideals returned by minAssGTZ when computed modulo 32003.

Theorem 2.6.3. For case (C3), system (2.74) is integrable if a22 = b22 and the coefficients ofthe systems satisfy one of conditions (α) − (η), (ι), (κ).

Proof. When condition (α) or (β) holds the corresponding system (2.74) is in the variety of theSibirsky ideal. In case (γ) the system (2.74) is written as

x = x− a40x5 + 2b04yx

4 + y2x3 − b04y4x− y5,

y = −y + x5 + a40yx4 − y3x2 − 2a40y

4x+ b04y5.

(2.77)

It admits the algebraic invariant curve of degree twelve

ℓ1 =1

8(x12 + 8a40yx

11 + 24a240y2x10 − 8b04y

2x10 + 32a340y3x9 − 48a40b04y

3x9

− 4y3x9 + 16a440y4x8 + 24b204y

4x8 − 24a40y4x8 − 96a240b04y

4x8 − 4b04x8 − 48a240y

5x7

+ 96a40b204y

5x7 − 64a340b04y5x7 + 24b04y

5x7 − 16a40b04yx7 − 12yx7 − 32a340y

6x6

− 32b304y6x6 + 96a240b

204y

6x6 + 96a40b04y6x6 + 6y6x6 + 16b204y

2x6 − 52a40y2x6

− 16a240b04y2x6 − 64a40b

304y

7x5 − 48b204y7x5 + 24a40y

7x5 + 96a240b04y7x5 − 64a240y

3x5

+ 32a40b204y

3x5 + 56b04y3x5 + 16b404y

8x4 + 24a240y8x4 − 96a40b

204y

8x4 − 24b04y8x4

− 16a340y4x4 − 16b304y

4x4 + 128a40b04y4x4 + 24y4x4 − 8a40x

4 + 32b304y9x3

− 48a40b04y9x3 − 4y9x3 − 64b204y

5x3 + 56a40y5x3 + 32a240b04y

5x3 + 32b04yx3

+ 24b204y10x2 − 8a40y

10x2 + 16a240y6x2 − 16a40b

204y

6x2

− 52b04y6x2 + 16a40b04y

2x2 + 36y2x2 + 8b04y11x− 16a40b04y

7x− 12y7x

+ 32a40y3x+ y12 − 4a40y

8 − 8b04y4 + 8)


yielding the integrating factor µ = ℓ−11 .

When condition (δ) holds, the system (2.74) is of the form

x = x− a40x5 − 2b04yx

4 − y2x3 − b04y4x− y5,

y = −y + x5 + a40yx4 + y3x2 + 2a40y

4x+ b04y5.

(2.78)

It can be transformed to (2.77) by the substitution x → αx, y → βy, where α = β5 andβ = (−1)1/12.

(ϵ) In this case the system has two invariant curves,

l1 = (4b04 − x4 − 4b04y2x2 − 4b204y

4)/

(4b04),

l2 = (16b404x8 + 2b04x

8 + 64b504y2x6 + 8b204y

2x6 + 64b604y4x4 + 16b304y

4x4

+y4x4 − 16b204x4 − 128b404yx

3 + 32b404y6x2 + 4b04y

6x2 + 64b304y2x2

−32b204y3x+ 32b504y

8 + 4b204y8 − 64b404y

4 + 32b304)/

(32b304),

which allow construction of the integrating factor µ = l121 l

−12 .

(ζ) This system can be transformed to system (ϵ) by the substitution x → αx, y → βy,where α = β5 and β = (−1)1/12. Thus it is integrable.

(η) In this case the system is Hamiltonian with the first integral

H(x, y) = (6xy − x6 − 6a40x5y − 2b22x

3y3 − 6b04xy5 − y6)/6

.(ι) When condition (ι) holds, system (2.74) is written as

x = x+x5

a31− a31yx

4 + y2x3 − y5,

y = −y + x5 − y3x2 +y4x

a31− a31y

5.

(2.79)

We found two invariant curves,

l1 = (x4 + 2a31y2x2 + a231y

4 + a31)/a31,

l2 = (a431x8 − 4a331yx

7 + 2a531y2x6 + 6a231y

2x6 − 8a431y3x5

−4a31y3x5 + a631y

4x4 + 12a331y4x4 + y4x4 + 4a231x

4 − 4a531y5x3

−8a231y5x3 − 8a431yx

3 + 6a431y6x2 + 2a31y

6x2 + 8a331y2x2

−4a331y7x− 8a231y

3x+ a231y8 + 4a431y

4 + 4a331)/

(4a331),

which yield the integrating factor

µ = l− 1

41 l−1

2 .

(κ) When condition (κ) holds, system (2.74) is written as

x = − x5

a31− a31yx

4 − y2x3 + x− y5,

y = x5 + y3x2 +y4x

a31+ a31y

5 − y.

(2.80)


It can be transformed into (2.79) by the change x → x(−1 − i)/√

2, y → y(1 + i)/√

2. Thus,(2.80) is integrable.

Note, that the remaining open case (θ) is the same as case (ζ) of case (C2).

Case (C4)

As above, using modular computations and then the rational reconstruction we found thefollowing components of the variety of the ideal I18 generated by the first 18 focus quantities ofsystem (2.74) with a04 = b40 and a22 = b22:(α) a240 + a40b04 + b204,−b31a40 + a13b04 = a13a40 + b31a40 + b31b04 = a213 + a13b31 + b231 = b13a40 +a31b04+b13b04 = a31a40−b13b04 = b13a13+a31b31+b13b31 = a31a13−b13b31 = a231+a31b13+b213 = 0;(β) a40 − b04 = a13 − b31 = a31 − b13 = 0;(γ) b22+ 1

2 = a240−a40b04+b204+a40+b04+1 = 83b04b

322− 5

3b04b22+b31 = 83a40b

322− 5

3a40b22+a13 =8a40b

322 − 5a40b22 + b13 = 8b04b

322 − 5b04b22 + a31 = 0;

(δ) b22− 12 = a240−a40b04+b204−a40−b04+1 = 8

3b04b322− 5

3b04b22+b31 = 83a40b

322− 5

3a40b22+a13 =8a40b

322 − 5a40b22 + b13 = 8b04b

322 − 5b04b22 + a31 = 0;

(ϵ) b22 + 12 = a40 + b04 − 1 = 8

3b04b322 − 5

3b04b22 + b31 = 83a40b

322 − 5

3a40b22 + a13 = 8a40b322 −

5a40b22 + b13 = 8b04b322 − 5b04b22 + a31 = 0;

(ζ) b22 − 12 = a40 + b04 + 1 = 8

3b04b322 − 5

3b04b22 + b31 = 83a40b

322 − 5

3a40b22 + a13 = 8a40b322 −

5a40b22 + b13 = 8b04b322 − 5b04b22 + a31 = 0;

(η) a40b04 − 1 = b13 + a40 = b22 + 1 = b31 = a13 = −b13b204 + a31 = 0;(θ) a40b04 − 1 = b13 − a40 = b22 − 1 = b31 = a13 = −b13b204 + a31 = 0;(ι) b13 − 1

2a13 = a31 − 12b31 = b04 = a40 = 0;

(κ)b22 = a40 + b04 = b31 + 1 = a13 + 1 = b13 = a31 = 0;(λ) a240 − a40b04 + b204 = b31b04 − a40 = b31a40 − a40 + b04 = b231 − b31 + 1 = b22 = a13 + b31 − 1 =b13 = a31 = 0.

Theorem 2.6.4. For case (C4), system (2.74) is integrable if and only if a22 = b22 and one ofconditions (α), (β), (η) - (λ) holds.

Proof. When condition (α) or (β) holds, the corresponding system (2.74) is time reversible.(η) Condition (η) is condition (ι) of Theorem 2.6.3.(θ) Condition (θ) is condition (κ) of Theorem 2.6.3.(ι) Condition (ι) is condition (β) of Theorem 2.6.1.(κ) Condition (κ) is condition (δ) of Theorem 2.6.2.(λ) Condition (λ) is condition (ϵ) of Theorem 2.6.2.

To summarize, we have found conditions for local integrability of systems (C1)–(C4), thatis, conditions of Theorem 2.6.1, conditions (α)–(η) of case (C2), (α)–(κ) of case (C3) and (α)–(λ) of case (C4). We believe that these conditions are the necessary and sufficient conditionsfor integrability of systems (C1)–(C4). However to prove this it remains to be shown that nocomponent was lost under the computations with modular arithmetic (that is, all Grobner basesarising in the radical membership test are 1) and to prove integrability of system correspondingto (η) of case (C2).

Chapter 3

Bifurcations of limit cycles

The study of perturbations of integrable systems is closely related to one of the most famousproblems in the qualitative theory of dynamical systems – Hilbert’s sixteenth problem on thenumber of limit cycles of two dimensional polynomial systems

x = Pn(x, y), y = Qn(x, y) (3.1)

(n is the maximum degree of the polynomials on the right-hand side of the system). In spite ofthe fact that Hilbert’s 16th problem was formulated more than a hundred years ago, it is notyet solved even for quadratic systems, that is, for system (3.1) with n = 2.

An essential part of the problem is the estimation of the maximum number of limit cycleswhich can bifurcate from a singular point of center or focus type under small perturbationsof coefficients of the system, the so-called cyclicity problem. This problem is also called thelocal Hilbert sixteenth problem. One of first major contributions to the study of the problemof cyclicity was the work of Bautin [5]. In this paper Bautin introduces the concept of cyclicityand solves the problem for quadratic systems. Other proofs of Bautin’s theorem are obtainedin [29, 35, 68, 70]. The general approach to the problem is described in books [56, 58]. Akey feature of the approach is that in the case of an elementary singular point the problemof cyclicity is reduced to the algebraic problem of searching for a basis of a polynomial idealgenerated by focus quantities, the Bautin ideal defined in the previous chapter. However, forsolving the latter problem there are no systematic approaches since the Bautin ideal is generatedby infinitely many polynomials defined implicitly with a recurrence formula. For this reason upto now the cyclicity problem has been solved only for systems in the form of a linear centerperturbed with homogeneous cubic nonlinearities [65] and for few special families of polynomialsystems.

Using algorithms of computational algebra in the case that the Bautin ideal is a radical idealthe problem can be solved in a relatively easy way (see [52]). The computational approach of [52]can be used for some systems but not in the case of generic polynomial systems. An approachwhich can be successfully applied for studying some systems with non-radical Bautin ideal hasbeen proposed recently in [39]. An efficient method which works in many cases is proposed byC. Christopher in [20].

In this chapter we describe the reduction of the cyclicity problem to the problem of computinga basis of the Bautin ideal. If the ideal is nonradical for the studied family of polynomial systemthen we introduce new variables which are connected to the Sibirsky ideal Isym (defined in

55

56 3 Bifurcations of limit cycles

Chapter 2) which describes the set of time-reversible systems. We expect that in the newvariables the Bautin ideal will be simpler and with the analysis of its structure we obtain theinformation about the upper bound for the cyclicity. Since the cyclicity of each component ofthe center variety is connected to its dimension, we compute the dimension of each componentand find the exact cyclicity of generic systems from the components.

3.1 The cyclicity problem 57

3.1 The cyclicity problem

In this chapter we consider systems of ordinary differential equations on R2 of the form

u = λu− v +n∑

j+k≥2

Aj,kujvk = P (u, v),

v = u+ λv +

n∑j+k≥2

Bj,kujvk = Q(u, v).

(3.2)

The degree of system (3.2) is n = maxdegP, degQ. Depending on nonlinear terms the originof system (3.2) is either a center (every orbit is an oval surrounding the origin), or a focus (everytrajectory spirals towards or away from the origin).

For system (3.2) we denote by (λ, (A,B)) the set of its parameters λ, Aj,k and Bj,k, and byE(λ, (A,B)) the associated space of parameters. Let also n(λ,(A,B)),ε denote the number of limitcycles of system (3.2) that lie wholly within an ε-neighborhood of the origin. Following Bautin[5] we give the following definition.

Definition 3.1.1. We say that the singularity at the origin for system (3.2) with fixed coefficients(λ∗, (A∗, B∗)) ∈ E(λ, (A,B)) has cyclicity k with respect to the space E(λ, (A,B)) if there existpositive constants δ0 and ε0 such that for every pair δ and ε satisfying 0 < δ < δ0 and 0 < ε < ε0

maxn(λ,(A,B)),ε : |(λ, (A,B)) − (λ∗, (A∗, B∗))|< δ = k.

In this section we show how the structure of the focus quantities can be used to move thecomputations to a different ring in order to get around the difficulty with nonradicality of theideal. Then, in the next section we apply this method to the family of cubic systems consideredin Section 2.4 and investigate the cyclicity of this system.

Following the complexification procedure described in the previous chapter we introduce thecomplex variable x = u+ iv, and obtain from (3.2) the equation

x = λx+ i(x−N∑

j+k=2

ajkxjxk). (3.3)

Equation (3.3) is just a complex form of real system (3.2). As usual adjoining to this equationits complex conjugate we obtain

x = λx+ ix−N∑

j+k=2

ajkxjxk, x = λx− ix+

N∑j+k=2

ajkxkxj ,

and replacing x by independent complex variable y and ajk by independent complex coefficientsbkj , we obtain

x = λx+ i(x−∑

j+k=2

ajkxjyk),

y = λy − i(y −N∑

j+k=2

bkjxkyj).

(3.4)


It is often convenient to use the focus quantities of system (3.4) to determine the cyclicityof the origin of system (3.2) when it is a center. If system (3.4) is a complexification of thecorresponding real system (3.2), then by the change of variables ajk = Ajk + iBjk and bkj =Ajk − iBjk we can obtain the focus quantities of the corresponding real system which we denotegRkk, that is, gRkk(A,B) = gkk(a(A,B), a(A,B)). Let A and B denote the vectors of parametersAj,k and Bj,k, respectively, of real system (3.2) and (A,B) = (A20, . . . , A0N , BN0, . . . , B20).

The next theorem reveals how the concept of minimality of the Bautin ideal is relatedto the cyclicity of the center at the origin. Given a Noetherian ring R and an ordered setV = v1, v2, . . . ⊂ R, we construct a basis MinBasis(I) of the ideal I = ⟨v1, v2, . . .⟩ as follows:

(a) initially set MinBasis(I) = vp, where vp is a first non-zero element of V;

(b) sequentially check successive elements vj , starting with j = p+1, adding vj to MinBasis(I)if and only if vj /∈ ⟨MinBasis(I)⟩.

The basis MinBasis(I) constructed as above is called the minimal basis of the ideal I with respectto the ordered set V. The cardinality of MinBasis(I) is called the Bautin depth of I [37]. Theproof of the following theorem can be found, for example, in [56] and [39].

Theorem 3.1.2 (Radical Ideal Cyclicity Bound). Suppose that for (3.4) with λ = 0 the followingtwo conditions hold:

(a) V(B) = V(BK),

(b) BK is a radical ideal.

Then the cyclicity of the origin of system (3.2) is at most the cardinality of MinBasis(BK), thatis the Bautin depth of B.

Conditions (a) and (b) of Theorem 3.1.2 imply that B = BK . To verify the first conditionwe just have to solve the center problem for corresponding system.

To verify condition (b) of Theorem 3.1.2 we find√

BK and then compute reduced Grobnerbases of

√BK and BK to check their equality. Another possibility to check that BK is a radical

ideal is to compute the primary decomposition of BK . By the Lasker-Noether DecompositionTheorem from Chapter 1 such a decomposition always exists. The output given by Singularis a list of pairs of ideals, where each ideal is specified by a list of generators. The first idealQj in each pair is a primary ideal in the primary decomposition of BK ; the second ideal Pj ineach pair is the associated prime ideal, that is, the radical of the first, Pj =

√Qj . If the second

ideal in each output pair is the same as the first one, which means that Qj =√Qj = Pj for all

j, then ideal Qi is prime. Thus, BK is a radical ideal.While there always exists K such that the first condition in Theorem 3.1.2 holds, the second

condition does not always hold. In some cases it is possible to overcome this difficulty causedby the nonradicality of the ideal BK . The idea is to move the problem to a different ring inwhich the image of the Bautin ideal becomes radical or has a simple structure. The method wasdeveloped in works of Levandovskyy, Logar, Romanovski and Shafer (see [38, 39]).

We use the following specific structure of the focus quantities which we briefly describe now.Fix the family (3.4), order the index set in some manner, and write it as S = (p1, q1), . . . , (pℓ, qℓ).Consistent with this order on S we order theN = 2ℓ coefficients as (ap1,q1 , · · · , apℓ,qℓ , bqℓ,pℓ , · · · , bq1,p1)so that any monomial appearing in gkk has the form

aν1p1,q1 · · · aνℓpℓ,qℓ

bνℓ+1qℓ,pℓ · · · bνNq1,p1

3.1 The cyclicity problem 59

for some ν = (ν1, . . . , νN ). Let L be the map from NN0 to NN

0 defined by (2.40). For ν =(ν1, . . . , νN ) ∈ NN

0 let [ν] denote the monomial in C[a, b] given by (3.12) and ν denote theinvolution of ν, ν = (νN , . . . , ν1).

Let M ⊂ NN0 be the monoid defined by (2.41) and Isym ⊂ C[a, b] be the symmetry or Sibirski

ideal defined by (2.42). Denote by G the reduced Grobner basis of Isym under any term order;then every element of G has the form [ν] − [ν] (see Subsection 2.3.1), where [ν] and [ν] have nocommon factors and a Hilbert basis of M is

MH = µ, µ : [µ] − [µ] ∪ ej + eN−j+1 : j = 1, . . . , ℓ and ± ([ej ] − [eN−j+1]) /∈ G, (3.5)

where ej = (0, . . . , 0, 1, 0, . . . , 0). Let M denote the cardinality of MH and for each νj ∈ MH lethj = [νj ] ∈ C[a, b]. By (2.43) the kth focus quantity of system (3.4) with λ = 0 has the form

gkk =∑

gαkkhα11 · · ·hαM

M . (3.6)

That is, the focus quantities gkk are elements of the subalgebra C[h1, . . . , hM ] generated by themonomials h1, . . . , hM .

The natural mapping

F : CN → CM : (a, b) 7→ (c1, . . . , cM ) = (h1(a, b), . . . , hM (a, b)),

induces the ring homomorphism

F c : C[c] → C[a, b]

:∑

cαcα11 · · · cαM

M 7→∑

cαhα11 (a, b) · · ·hαM

M (a, b).

In some cases (see e.g. [39]) the ideal which is non-radical in C[a, b] becomes radical in C[h1, . . . , hm].If the Bautin ideal is non-radical also in the corresponding subalgebra, but has a primary

decomposition of the form ∩si=1Qi, where for 1 ≤ i ≤ k

√Qi = Qi while

√Qi = Qi for

i = k + 1, . . . , s, then the following statement can be helpful.

Proposition 3.1.3. Let I = ⟨g1, . . . , gt⟩ be an ideal in C[x1, . . . , xn] such that the primarydecomposition of I is given as I = P1∩· · ·∩Pk∩Q1∩· · ·∩Qm, where Pi and Qi are primary idealssuch that Pi =

√Pi for i = 1, . . . , k, and Qj =

√Qj for j = 1, . . . ,m. Let Q = Q1∩· · ·∩Qm and

g be a polynomial vanishing on V(I). Let x∗ = (x∗1, . . . , x∗n) be an arbitrary point of V(I)\V(Q).

Then in a small neighborhood of x∗ we have g = g1f1 + · · · + gtft, where f1, . . . , ft are powerseries convergent at x∗.

Proof. By our assumption√I = P1 ∩ · · · ∩ Pk ∩

√Q1 ∩ · · · ∩

√Qm.

Let g ∈√I. Then g ∈ P1 ∩ · · · ∩ Pk and g ∈

√Q. For any polynomial q ∈ Q we have

qg ∈ P1 ∩ · · · ∩ Pk and qg ∈ Q, hence qg ∈ I; in particular there exist f1, . . . , ft ∈ C[x1, . . . , xn]such that

qg = f1g1 + · · · + ftgt. (3.7)

Since x∗ = (x∗1, . . . , x∗n) ∈ V(I) \V(Q), there exists q ∈ Q such that q(x∗) = 0. Since such q is

invertible in the local ring at x∗, we can write:

g =f1qg1 + · · · +

ftqgt. (3.8)

Clearly, for any l = 1, . . . , t we can express flq as a power series in a neighborhood of x∗.


A germ of an analytic function at a point θ∗ ∈ kn (k is R or C) is an equivalence class ofanalytic functions under the relation f is equivalent to g if there is a neighborhood of θ∗ onwhich f and g agree. We denote by Gθ∗ the ring of germs of analytic functions of θ at the pointθ∗ ∈ kn.

The next theorem, whose proof can be found in [56], provides a tool for determining theupper bound of the cyclicity for systems of the form (3.2).

Theorem 3.1.4. Suppose that for (A∗, B∗) ∈ E(A,B), the minimal basis M with respect to theretention condition of the ideal J = ⟨gR11, gR22, . . .⟩ in G(A∗,B∗) for the corresponding system of theform (3.2) with λ = 0 consists of m polynomials. Then the cyclicity of the origin of the systemof the form (3.2) that corresponds to the parameter string (0, (A∗, B∗)) ∈ E(λ, (A,B)) is at mostm.

3.2 The cyclicity of the subfamily of cubic systems 61

3.2 The cyclicity of the subfamily of cubic systems

We apply the approach discussed in the previous section to the study of the cyclicity problemfor a family of real cubic systems whose expression in the complex form is

x = λx+ ix(1 − a10x− a20x2 − a11xx− a02x

2), (3.9)

where x = u + iv. The motivation for studying system (3.9) is that it is one of very few 9-parameter cubic systems (we say 9-parameter because each complex parameter aij depends ontwo real parameters) where the computation of the primary decomposition of the Bautin idealis feasible (because of computational complexity it is extremely difficult to treat 9-parametercubic systems with modern tools of computational algebra even using very powerful computers).

If we add to (3.9) the complex conjugate equation and consider x as a new unknown functiony, and aij as new parameters bji we obtain the associated complex system

x = λx+ ix(1 − a10x− a20x2 − a11xy − a02y

2),

y = λy − iy(1 − b01y − b02y2 − b11xy − b20x

2).(3.10)

To obtain a bound for the cyclicity of the origin of the real system (3.9) we shall study in detailthe structure of the Bautin ideal of system (3.10) with λ = 0 as well as its variety. We will seethat the approach from Section 3.1 gives an estimation for the cyclicity of an elementary center orfocus for “almost all” points of the center variety. The solution of the center problem for system(3.10) with λ = 0 is established in Theorem 2.4.1 where we have proven that V(B) = V(B4).Using this preliminary result we prove the next theorem providing a bound for the cyclicity ofthe origin of system (3.9).

Theorem 3.2.1. Assume that |a20|+|a11|+|a02|= 0. Then the cyclicity of the center at theorigin of system (3.9) is at most four.

For our proof we need the following lemma.

Lemma 3.2.2. The focus quantities gkk of system (3.10) belong to the polynomial subalgebraC[h1, . . . , h12] ⊂ C[a, b], where

h1 = a11 h2 = b11 h3 = a20a02 h4 = b20b02h5 = a210b02 h6 = a20b

201 h7 = a210a02 h8 = b20b

201

h9 = a10b01 h10 = a20b02 h11 = a11b11 h12 = a02b20.(3.11)

Proof. The set of indexes of coefficients of our system is S := (1, 0), (2, 0), (1, 1), (0, 2). Con-sistently with this we order the eight coefficients as (a10, a20, a11, a02, b20, b11, b02, b01) so that anymonomial, denoted by [ν], appearing in gkk has the form

[ν] := aν110aν220a

ν311a

ν402b

ν520b

ν611b

ν702b

ν801 (3.12)

for some (ν1, . . . , ν8) ∈ N80. Let L : N8

0 → Z2 be the map defined by

L(ν) = ν1(1, 0) + ν2(2, 0) + ν3(1, 1) + ν4(0, 2)

+ ν5(2, 0) + ν6(1, 1) + ν7(0, 2) + ν8(0, 1)(3.13)


and

M = ν ∈ N80 : L(ν) = (k, k), k = 0, 1, 2, . . ..

By ν we denote the involution of ν, that is ν = (ν8, . . . , ν1). By Proposition 2.3.2 the focusquantities of system (3.10) with λ = 0 have the form

gkk =∑

ν:L(ν)=(k,k)

g(ν)([ν] − [ν]) (3.14)

where L is the map defined by (3.13).Using the algorithm presented in Subsection 2.3.1 we compute the Hilbert basis of M ,

obtaining the 12-element set

(0010 0000) (0200 0001) (0100 0010)(0000 0100) (1000 0020) (1000 0001)(1001 0000) (1001 0000) (0010 0100)(0000 1001) (0000 1020) (0001 1000).

We denote by νj the jth element of this list and let hj = [νj ] ∈ C[a, b], where νj is defined by(3.12). Thus,

gkk =∑

g(α)kk h

α11 · · ·hα12

12 ,

where g(α)kk ∈ C, k ∈ N, that is, the focus quantities gkk of system (3.10) with λ = 0 belong to

the polynomial subalgebra C[h1, . . . , h12], which proves our lemma.

To prove our cyclicity bound we first attempt to use Theorem 3.1.2. The equality V(B) =V(B4) was established in Theorem 2.4.1. Thus condition (a) of Theorem 3.1.2 holds. Now weshow that B4 is not radical and, therefore, the condition (b) (of Theorem 3.1.2) is not fulfilled.To see this we compute the generators of

√B4 using the command radical of singular [34].

Now we compute the reduced Grobner bases of the ideals B4 and√

B4, respectively. It is clearthat B4 is radical if and only if B4 and

√B4 are the same. It turns out that (we checked this

using the command reduce of singular) some of the elements of√

B4 do not reduce to zeromodulo a Grobner basis of B4. Therefore

√B4 * B4.

However we use the following approach. By Lemma 3.2.2 focus quantities of system (3.10)belong to the polynomial subalgebra C[h1, . . . , h12], where hj = hj(a, b) for j = 1, . . . , 12 aregiven in (3.11). We define the polynomial mapping

F : C8 → C12 : (a, b) 7→ (c1, . . . , c12) = (h1(a, b), . . . , h12(a, b)),

which induces the C-algebra homomorphism

F ∗ : C[c] → C[a, b] :∑

d(α)cα11 · · · cα12

12 7→∑

d(α)hα11 (a, b) · · ·hα12

12 (a, b),

where d(α) ∈ C.

Now we shall compute preimages of g11, . . . , g44 under the map F ∗, which we denote bygF11, . . . , g

F44, that is, we seek to express the focus quantities g11, . . . , g44 ∈ C[a, b] defined by

(2.51) in the new variables c1, . . . , c12. Since the focus quantities can be lengthy polynomials, weuse the following algorithmic procedure (whose correctness is proved in [23]) for finding those


expressions. Choose any elimination ordering with a, b > c: for instance, the lexicographicalorder with a20 > a10 > a11 > a02 > b20 > b11 > b01 > b02 > c1 > · · · > c12. Let c = (c1, . . . , c12)and denote by J ⊂ C[a, b, c] the ideal generated by cj − hj(a, b), that is J = ⟨cj − hj(a, b) :j = 1, . . . , 12⟩. Computing a Grobner basis JG of the ideal J in C[a, b, c] with respect to theelimination ordering as above and forming the Grobner basis R = JG ∩C[c] of the ideal J ∩C[c]yields a 10-element set r1, . . . , r10 given by

c3c8 − c6c12 c4c7 − c5c12 c4c6 − c8c10 c3c5 − c7c10c3c4 − c10c12 c1c2 − c11 c29c12 − c7c8 c29c10 − c5c6c4c

29 − c5c8 c3c

29 − c6c7.

By the Elimination Theorem the set given above is a Grobner basis of R. Finally, the divisionof each g11, . . . , g44 by JG yields polynomials gF11, . . . , g

F44:

gF11 = −c1 + c2;gF22 = 6c1c9 − 6c2c9 − c3 + c4;gF33 = 12c21c9 − 12c22c9 − 54c1c

29 + 54c2c

29 − 7/2c1c3 + 5/2c2c3 − 5/2c1c4 + 7/2c2c4 − 3/2c1c5 + 2c2c5 −

2c1c6 +3/2c2c6−15/2c1c7 +8c2c7−8c1c8 +15/2c2c8 +10c3c9−10c4c9 +2c1c10−2c2c10 +4c1c12−4c2c12;

gF44 = 120c31c9 − 120c32c9 − 300c21c29 + 300c22c

29 + 740c1c

39 − 740c2c

39 − 27/4c21c3 + 13/4c22c3 − 13/4c21c4 +

27/4c22c4+36c21c5+44c22c5−44c21c6−36c22c6−54c21c7−16c22c7+16c21c8+54c22c8+958/9c1c3c9−517/9c2c3c9+

517/9c1c4c9−958/9c2c4c9 + 38c1c5c9−139/3c2c5c9 + 139/3c1c6c9−38c2c6c9 + 198c1c7c9−619/3c2c7c9 +

619/3c1c8c9−198c2c8c9+15/4c21c10−15/4c22c10−353/9c1c9c10+353/9c2c9c10−300c1c9c11+300c2c9c11+

25/4c21c12 − 25/4c22c12 − 332/9c1c9c12 + 332/9c2c9c12− 9/2c23 + 9/2c24 + 15/2c4c5 − 15/2c3c6 − 25/6c3c7 −132c6c7+25/6c4c8+132c5c8+3c3c10−3c4c10−43/6c7c10+43/6c8c10+5/2c3c11−5/2c4c11−157/2c5c11+

157/2c6c11 + 143/2c7c11 − 143/2c8c11 + 5c3c12 − 5c4c12 + 29/6c5c12 − 29/6c6c12 + 1/3c7c12 − 1/3c8c12.

Geometrically equations (3.11) define a graph in C8+12 and the projection of the graph toC12 denoted by W is computed using the Elimination Theorem. The Zariski closure of theprojection is the variety of the ideal R, W = V(R). Hence, to show that

Vc = V(Nc), where Nc = (B4 + J) ∩ C[c],

we compute the reduced Grobner basis of the ideal Nc, and find that it is the same as thereduced Grobner basis of the ideal ⟨gF11, . . . , gF44, R⟩ ⊂ C[c]. Therefore Nc = ⟨gF11, . . . , gF44, R⟩,which in turn, gives

Vc = V(Nc) = V(⟨gF11, . . . , gF44, r1, . . . , r10⟩).By the natural isomorphism of C[W ] with C[c]/R the ideal ⟨gF11, . . . , gF44⟩ is radical if and

only if the ideal ⟨gF11 + R, . . . , gF44 + R⟩ in C[c]/R is radical. It is easy to see that this is true ifthe ideal

K = ⟨gF11, . . . , gF44, r1, . . . , r10⟩ (3.15)

is radical in C[c].Using the routine primdecGTZ of Singular we compute the primary decomposition of K

and obtainK = P1 ∩ · · · ∩ P5 ∩Q, (3.16)

where P1, . . . , P5 are prime ideals but unfortunately Q is not prime. The output of calculationsin Singular is given in Appendix D. Thus K is not a radical ideal in C[c]/R. Therefore, themethod of [39] cannot be directly applied, since condition (b) of Theorem 3.1.2 does not holdeven in C[c].


Proof of Theorem 3.2.1. We note, that the ideal Q in (3.16) is a primary ideal such that√Q =

⟨c11, c10−c12, c6 +c8, c5 +c7, c4 +c12, c3 +c12, c2, c1, c29c12−c7c8⟩. Thus K has the structure as in

Proposition 3.1.3. In the parameter space E(a, b), the variety V(Q) is defined by a20b201+b20b

201 =

a210a02 + a210b02 = a02b20 + b20b02 = b11 = a11 = a20b02 + b20b02 = a20a02 − b20b02 = 0. Theintersection V(Q) with the parameter space E(a) is the set a20 = a11 = a02 = 0. Let now(a∗, a∗) be a point from E(a, b) corresponding to system (3.10). If |a20|+|a11|+|a02|= 0 thenF (a∗, a∗) /∈ V(Q). Therefore, by Proposition 3.1.3, there exist rational functions fj,k, sj,k suchthat for c ∈W in a neighborhood of F (a∗, a∗) with |a20|+|a11|+|a02|= 0 we have

gFkk = gF11f1,k + · · · + gF44f4,k +

10∑j=1

rjsj,k. (3.17)

Applying F ∗ to (3.17) we get that

gkk = g11f1,k + g22f2,k + g33f3,k + g44f4,k

holds for all k > 4 and some rational functions fi,k (i = 1, . . . , 4) in a neighborhood of (a∗, a∗).Thus, by Theorem 3.1.4 the cyclicity of the center at the origin of system (3.9) with

|a20|+|a11|+|a02|= 0 is at most four.

3.2.1 Cyclicity of the components

In this section we study the number of limit cycles that can bifurcate from each component ofthe center variety of system (3.9). Our approach is based on a result of [20], which relates thesenumbers to the dimension of each component.

First we shall introduce some notation. Denote by gRkk the polynomials obtained by replacingin gkk the variable bqp with apq. Then the center variety of the real system (3.9) is the variety VR

in E(a) of the ideal BR = ⟨gR11, gR22, . . .⟩. We denote by Jp(BRk ) the Jacobian of the polynomials

gR11, gR22, . . . , g

Rkk evaluated at the point p and by rankJk

p the rank of Jp(BRk ).

Theorem 3.2.3 ([20]). Assume that for system (3.3) with λ = 0 and a point p ∈ VR, rankJkp =

k. Then the codimension of VR is at least k and there are bifurcations of (3.3), which producelocally k limit cycles from the center corresponding to the parameter value p.

Moreover, if p lies on a component C of VR of codimension k, then p is a smooth point ofthe center variety, and the cyclicity of p and any generic point of C is exactly k.

In order to apply this theorem, first we find all irreducible components of VR. This canbe obtained from the components of the complex center variety of system (3.10) described inTheorem 2.4.1 by setting

a10 = A10 + iB10 a20 = A20 + iB20

a11 = A11 + iB11 a02 = A02 + iB02

b02 = A20 − iB20 b01 = A10 − iB10

b11 = A11 − iB11 b20 = A02 − iB02.

(3.18)

The capital letters indicate the coefficients of the real system written in the complex form as(3.3). Note that coefficients appearing in (3.18) are not the coefficients appearing in (3.2). Thecoefficients of (3.2) are some elaborate expressions of coefficients of (3.18).

The following theorem describes in details the variety VR.


Theorem 3.2.4. The center variety in R8 of the real system (3.9) with λ = 0 consists of thefollowing three irreducible components

VR = V(G1) ∪V(G2) ∪V(G3),

whereG1 = ⟨B11, A20B02 +A02B20, A

210B02 + 2A02A10B10 −B02B

210, A

210B20

− 2A10A20B10 −B210B20⟩

G2 = ⟨B11, A02 +A20, B02 −B20⟩G3 = ⟨A11, B11, A02, B02⟩,

of dimension 5, 5, and 4, respectively.

Proof. We obtain the idealsG1, G2 andG3 by applying the change of variables (3.18) respectivelyto the generators of the ideals J1, J2 and J5 of Theorem 2.4.1. We note that applying (3.18)to J3 and J4 of Theorem 2.4.1 results in the same ideal ⟨A11, B11, A02, B02, A20, B20⟩, which is asubvariety of G3.

It is straightforward to see that the dimensions of V(G2) and V(G3) are 5 and 4, respectively.Now we show that dimV(G1) = 5.

We begin by finding a rational parametrization of V(G1). We claim that one is given by

A11 = f1 A02 = f2 A10 = f3 B10 = f4

A20 = f5 B11 = f0 B02 =f6g1

B20 =f7g1,

(3.19)

where f0=0, f1= t, f2=u1, f3=u2, f4=u3, f5=u4, f6=−2u1u2u3, f7=2u2u3u4 and g1=u22−u23.To prove our claim we construct an ideal Ie by eliminating the variables w, t, u1, u2, u3, u4 fromthe ideal

⟨1 − wg1, A11 − f1, A02 − f2, A10 − f3, B10 − f4, A20 − f5, g1B02 − f6, g1B20 − f7⟩

in the ring

R[w, t, u1, u2, u3, u4, A02, A10, A11, A20, B20, B10, B02].

Computations show that Ie = G1. Therefore, by Theorem 1.2.17 we get that (3.19) is a rationalparametrization of G1 and dimV(G1) ≤ 5. Now, since the rank of the Jacobian of f0, . . . , f7 atthe point t = 0, u1 = 1, u2 = −1, u3 = 2, u4 = 3 is five, we obtain that, in fact, dimV(G1) = 5,which completes the proof of our theorem.

We define the following polynomials:

F1 = A11(B02 −B20)(A210 +B2

10)

F2 = −A11(2A02A10B10 + (B210 −A2

10)B20)

F3 = (2A10A20B10 −A210B20 +B2

10B20)2(A2

20 +B220).

Theorem 3.2.5. The cyclicity of a generic point p of V(G1) with F1(p) = 0 and of a point p′

of V(G2) with F2(p′) = 0 is three. The cyclicity of a point p′′ of V(G3) with F3(p

′′) = 0 is four.


Proof. We find that for V(G1), rank J3p = 3 at p with F1(p) = 0 and similarly for V(G2) we

obtain rank J3p′ = 3 at the point p′ with F2(p

′) = 0. Therefore, by Theorem 3.2.3 three limitcycles bifurcate from the origin for the systems corresponding to p and p′, respectively. ForV(G3) we have rank J4

p′′ = 4 at the point p′′ with F3(p′′) = 0. Since the codimension of V(G3)

is 4, by Theorem 3.2.3 the cyclicity for the system corresponding to p′′ is 4.

Chapter 4

Bifurcations of critical periods

If some system has a center then the next question is whether the center is isochronous. Bydefinition, if all periodic solutions in a neighborhood of the center have the same period, then thecenter is isochronous. The isochronicity problem is to find conditions on the parameters of thesystem for which the center is isochronous. Isochronicity was studied already in the 17th centurywhen Christian Huygens observed that a pendulum clock has a monotone period function andoscillates with a shorter oscillation period when it has a smaller energy, that is, when the clock’sspring unwinds. He wanted to make a clock which would oscillate isochronously and would bemore precise. It appears, his solution, the cycloidal pendulum, is probably the first exampleof a nonlinear isochrone. The problem attracted attention again in the early sixties of the lastcentury when Japanese mathematician Minoru Urabe studied isochronicity of some Hamiltoniansystems (see [66]). From this time many studies have been devoted to the problem and it wasresolved for some classes of planar systems.

A survey on the extensive study on the isochronicity problem was given by J. Chavarrigaand M. Sabatini in [16]. We mention that the conditions for which the origin is an isochronouscenter are obtained in the case of a linear center with homogeneous perturbation with quadraticpolynomials [44], with cubic polynomials [22, 48, 62], with polynomials of degree five [54] andfor some specific families of polynomial systems.

Another problem closely related to the problem of the center and to the isochronicity andcyclicity problems is the problem of bifurcations of critical periods. The theory of bifurcations ofcritical periods started with the work of C. Chicone and M. Jacobs [18]. They applied the theorydeveloped to the study of quadratic systems and some Hamiltonian systems. Bifurcations ofcritical periods of the so-called Kukles system are studied in [61], of the system with homogeneouscubic nonlinearities in [60], of time-reversible cubic systems in [69] and cubic Lienard system in[71].

In this chapter we assume that the singularity of the real system under consideration isknown to be a center. We first present a method for determining whether or not the centeris isochronous, i.e. whether every periodic orbit in a neighborhood of the origin has the sameperiod. For this purpose we describe the period function. We present an approach for computingthe period function in polar coordinates and another one which is based on normal form theory.Then in the last section we apply described approaches for solving the isochronicity problemand the problem of critical periods for some families of three-dimensional cubic systems.

67

68 4 Bifurcations of critical periods

4.1 The period function

We consider the polynomial system of the form

u = −v + Pn(u, v), v = u+Qn(u, v), (4.1)

where Pn and Qn are polynomials of degree at most n without constant and linear terms. Thecoefficients of Pn and Qn are the parameters. Introducing the polar coordinates u = r cosφ,v = r sinφ yields the equation of the trajectories

dr

dφ=

r2F (r, cosφ, sinφ)

1 + rG (r, cosφ, sinφ)= R (r, φ) . (4.2)

As in Section 2.1 we can choose the line segment Σ = (u, v) ; v = 0, 0 ≤ u ≤ r∗, where r∗

is chosen to be small enough.

Now we consider the solution of (4.2) with the initial condition r = r0, φ0 = 0, and expandit in a power series in r0 to obtain

r (φ, r0) = w1 (φ) r0 + w2 (φ) r20 + w3 (φ) r30 + · · · , (4.3)

which is convergent for all φ ∈ [0, 2π] and all |r0| ≤ r∗. The r (φ) from (4.3) is a solution of (2.5)and inserting r (φ, r0) into (2.5) yields recurrence differential equations (2.8) for the functionswj (φ) . We consider one revolution of r = r (φ, r0) beginning on r0 ∈ Σ where φ is assumed tobe 0 and study the return to Σ (which happens at φ = 2π, that is, after one revolution). Thusthe Poncare return map R(r) is defined by

R (r0) = r (2π, r0) = r0 + w2 (2π) r20 + w3 (2π) r30 + · · · , (4.4)

where the coefficients ηj := wj (2π) for j > 1 of (4.4) are the Lyapunov numbers. Zeros of thedifference function P (r0) = R(r0) − r0 correspond to closed orbits.

Suppose now that the origin of (4.1) is a center. Then for small enough r0 < r∗ the trajectoryof (4.1) is an oval (i.e. a simple closed curve) and we can consider the so-called period functionT (r). To obtain T (r) we note that

φ = 1 + rG(r, cosφ, sinφ) = 1 +

∞∑k=1

ζk(φ)rk.

Since w1 ≡ 0 substituting instead of r expression (4.3) with r0 = r into this equation we obtain

φ = 1 +

∞∑k=1

ζk (φ)

(r +

∞∑k=2

wk (φ) rk

)k

.

Therefore,

dt

dφ=

1

1 +∞∑k=1

Fk (φ) rk= 1 +

∞∑k=1

Ψk (φ) rk,

4.1 The period function 69

where∞∑k=1

Ψk (φ) rk is an analytic function. After integrating we obtain

t = φ+

∞∑k=1

∫ φ

0Ψk (s) ds · rk = φ+

∞∑k=1

θk (φ) rk

(note that r∗ is small enough and actually fixed, so the series above converges for φ ∈ [0, 2π])and setting φ = 2π (and t = T ) one can obtain the ”one revolution time” or the period

T (r) = 2π

(1 +

∞∑k=1

Tkrk

), (4.5)

where Tk = 12πθk (2π) for k ≥ 1. A center is isochronous if all solutions in its neighborhood

have the same period. Thus the center at the origin of system (4.1) is isochronous if and only ifTk = 0 for k ≥ 1.

Definition 4.1.1. Assume that T (r) ≡ 0. Any value 0 < r < r∗ for which T ′(r) = 0 is called acritical period.

The question of interest is the maximum number of zeros (critical periods) of T ′ that canbifurcate from the zero at r = 0, when the coefficient string a of system (4.1), which we writein the complex form

x = i(x−n−1∑

p+q=1

apqxp+1xq), (4.6)

is perturbed from a∗ but remains within the center variety V RC of family (4.6).

Let E be a subset of Rn and let F : R× E → R : (z, θ) 7→ F (z, θ) be an analytic function,which in the neighborhood of z = 0 is written as

F (z, θ) =

∞∑j=0

fj(θ)zj , (4.7)

where for j ∈ N0, fj(θ) is an analytic function and the series (15) is convergent in a neighborhoodof (z, θ) for any θ∗ ∈ E . According to Definition 6.1.1 in [56] if θ∗ ∈ E is such that F (z, θ∗) = 0and nz,ε denote the number of isolated zeros of F (z, θ) in the interval 0 < z < ε, then we saythat the multiplicity of F (z, θ) at θ∗ with respect to the set E is equal to m if there exist δ0 > 0and ε0 > 0, such that for every 0 < ε < ε0 and 0 < δ < δ0

maxθ∈Uδ(θ∗)∩E

nθ,ε = m.

We see that the problem of cyclicity is the problem of the multiplicity of the difference func-tion P(r) and the problem of bifurcations of critical periods is the problem of the multiplicityof the function T ′, where T (r) is the period function and E is not the full space Rn2+3n but thecenter variety V R

C of family (4.6).According to results of [56, Chapter 6] if ⟨T1, T2, T3, . . .⟩ = ⟨T2, T4, . . . , T2k⟩, then at most

k − 1 critical periods bifurcate from any center of system (4.1) after small perturbations.


4.2 An approach to studying isochronicity and bifurcations ofcritical periods

In the previous section we described the isochronicity problem and the problem of critical periodbifurcations for a real polynomial planar system using the period function obtained introducingpolar coordinates. Since such an approach demands integration of trigonometric functions, whichcan be a very difficult computational problem, we describe another approach based on normalforms. We first describe the isochronicity problem for a real system (4.1) and its complexificationand then we describe the linearizability problem for more general complex systems.

4.2.1 The isochronicity problem

Let the system

u = −v +

n∑p+q=2

αpqupvq,

v = u+

n∑p+q=2

βpqupvq

(4.8)

have a center at the origin (all solutions are periodic).By a result of Poincare and Lyapunov (see e.g. Theorem 4.2.1 of [56] for the details) the

origin is an isochronous center for system (4.1) if and only if there is an analytic change ofcoordinates

u = z + ϕ(z, w), v = w + ψ(z, w)

that reduces (4.1) to the canonical linear center z = −w, w = z. This result tells us that theisochronicity of a planar analytic system is equivalent to its linearizability. Therefore, insteadof studying the isochronicity of planar polynomial systems, we can study their linearizability.Since linearizability has natural generalization to the complex setting, we can transform ourreal system (4.1) to a complex system on C2 and study the complex varieties. If we apply thecomplexification procedure described at the beginning of Section 2.2, we first obtain system(4.6) and then the system

x1 =i(x1 −n∑

p+q=2

apqxp+11 xq2) = P (x1, x2),

x2 = − i(x2 −n∑

p+q=2

bqpxq1x

p+12 ) = Q(x1, x2),

(4.9)

where x1 and x2 are complex variables. The resonant pairs for (4.9) are (1, (k + 1, k)) and(2, (k, k + 1)), k ∈ N, so when we apply normalizing transformation (2.19), namely

x1 = y1 +∑

p+q≥2

h(j,k)1 yj1y

k2 , x2 = y2 +

∑p+q≥2

h(j,k)2 yj1y

k2 (4.10)

to reduce system (4.9) to the normal form, we obtain

y1 = y1(i+ Y1(y1y2)), y2 = y2(−i+ Y2(y1y2)), (4.11)

4.2 An approach to studying isochronicity and bifurcations of critical periods 71

where

Y1(y1y2) =

∞∑j=1

Y(j+1,j)1 (y1y2)

j and Y2(y1y2) =

∞∑j=1

Y(j,j+1)2 (y1y2)

j . (4.12)

System (4.9) is linearized by the transformation (4.10) when (4.11) reduces to

y1 = iy1, y2 = −iy2.

This is the case if and only if all coefficients Y(k+1,k)1 , Y

(k,k+1)2 are equal to zero. We have

observed already that if we consider the isochronicity problem and the linearizability problemfor the complexification (4.9) of real system (4.1), these two problems are equivalent, therefore,we can say that the real system corresponding to system (4.9) has an isochronous center if

and only if Y(k+1,k)1 = Y

(k,k+1)2 = 0, k ∈ N . Coefficients Y

(k+1,k)1 , Y

(k,k+1)2 are called normal

form coefficients. Clearly there is no geometrical meaning in the term “isochronous center” forthe complexification of a real system. But since it is easier to deal with complex varieties wecan obtain conditions of linearizability of system (4.9) and then going back to the original realsystem (4.1) we preserve this information and obtain conditions for isochronicity of real system(4.1).

To obtain information about critical periods we compute the period function T (r) in thecomplex setting. The first equation in the normal form (4.11) is given by

y1 = y1(i+1

2[G(y1y2) +H(y1y2)]), (4.13)

where G and H are functions defined by (2.20). By Theorem 2.2.2 the origin is a center if andonly if in the normal form (4.13) G ≡ 0, in which case H has purely imaginary coefficients. Nowwe define H by

H(w) = −1

2iH(w).

We can consider (4.11) as a complexification of a real system (see Proposition 3.2.1 in [56]). So,we obtain the description of the real system (4.1) in complex form by replacing every y2 by y1in each equation of (4.11) and then setting y1 = reiφ which yields the system

r = 0, φ = 1 + H(r2). (4.14)

Now we integrate the expression for φ in (4.14) and obtain the expression for period function of(4.9)

T (r) =2π

1 + H(r2)= 2π(1 +

∞∑k=1

p2kr2k). (4.15)

Recall from (2.21) that H(w) =∑∞

k=1H2k+1wk and therefore H(w) =

∑∞k=1 H2k+1w

k. Thecenter is isochronous if and only if p2k = 0 for k ∈ N or, equivalently, H2k+1 = 0 for k ∈ N. Thecoefficient p2k is called the kth isochronicity quantity.

To derive formulas for computing p2k we invert the series on the left-hand side of (4.15).Writing (1 +

∑∞k=1 akx

k)−1 = 1 +∑∞

k=1 bkxk and clearing the denominators we obtain 1 =

1+∑∞

k=1(ak +ak−1b1+ · · ·+bk)xk. Since all coefficients in the sum on the right-hand side of lastequations must be zero we obtain b1 = −a1 and for k ≥ 2 bk = −a1bk−1−a2bk−2−· · ·−ak−1b1−ak.We see, that the coefficients bk can be recursively computed in terms of the ak. In our case, by


(4.15) ak = H2k+1 = −12 iH2k+1 = −1

2 i(Y(k+1,k)1 −Y (k,k+1)

2 ) and bk = p2k. Thus, the polynomials

p2k and H2k+1 satisfy

p2 = −H3 and p2k = −H2k+1 mod ⟨H3, . . . , H2k−1⟩ for k ≥ 2. (4.16)

From (4.16) we see that ⟨H2k+1 : k ∈ N⟩ = ⟨p2k : k ∈ N⟩ and then we can obtain explicitexpressions for the polynomials p2k. First three are

p2 =i

2(Y

(2,1)1 − Y

(1,2)2 ),

p4 =i

2(Y

(3,2)1 − Y

(2,3)2 ) − 1

4(Y

(2,1)1 − Y

(1,2)2 )2,

p6 =i

2(Y

(4,3)1 − Y

(3,4)2 ) − 1

2(Y

(2,1)1 − Y

(1,2)2 )(Y

(3,2)1 − Y

(2,3)2 )

− i

8(Y

(2,1)1 − Y

(1,2)2 )3.

(4.17)

To investigate the multiplicity of the function

T ′(r, (a, a)) =∞∑k=1

2kp2k(a, a)r2k−1,

we need to analyze the ideal ⟨2kp2k : k ∈ N⟩ = ⟨p2k : k ∈ N⟩ ⊂ C[a, b].Let P = ⟨p2k : k ∈ N⟩ ⊂ C[a, b] denote the ideal generated by isochronicity quantities and

PK = ⟨p2, . . . , p2k⟩ the ideal generated by the first k isochronicity quantities.Let H = ⟨H2j+1 : j ∈ N⟩ = ⟨H2j+1 : j ∈ N⟩ and by VI we denote isochronicity variety, that

is, VI = V(H )∩VC . By Proposition 4.2.7 in [56] VI = VL , where VL = V(Y ) is linearizabilityvariety of the system (4.9), that is the variety of the ideal Y , where the ideal Y is defined as

Y := ⟨Y (j+1,j)1 , Y

(j,j+1)2 : j ∈ N⟩ ⊂ C[a, b].

For any k ∈ N we set Yk = ⟨Y (j+1,j)1 , Y

(j,j+1)2 : j = 1, . . . , k⟩.

Implementing the Normal Form Algorithm (see Table 2.1 in [56]) in a computer algebrasystem it is possible to derive the normal form of (4.9) through a low order, and then to obtainexpressions for p2k using (4.17).

Although the isochronicity quantities p2k are defined for all choices of the coefficients (a, b),they are relevant only for (a, b) ∈ VC , the center variety, where their vanishing identifiesisochronicity for real center and linearizability for all centers.

4.2.2 The linearizability problem

In order to find all systems (4.9) which are linearizable by a convergent transformation of theform (4.10) we can construct a normal form of (4.9) by means of a transformation (4.10).However from the computational point of view it is more efficient to look for the inverse of(4.10).

The linearizability problem for a system of the form

x = x−n−1∑

p+q=1

ap,qxp+1yq, y = −y +

n−1∑p+q=1

bq,pxqyp+1 (4.18)

4.2 An approach to studying isochronicity and bifurcations of critical periods 73

is to decide whether the system can be transformed to the linear system z = z, w = −w bymeans of a formal change of the plane variables which can be of the form (4.10) or of the form

z = x+

∞∑m+j=2

u(1)m−1,j(a, b)x

myj ,

w = y +∞∑

m+j=2

u(2)m,j−1(a, b)x

myj .

(4.19)

Differentiating with respect to t on both sides of the two equalities in (4.19) and substituting(4.18) in the resulting equalities and then using (4.19) and (4.18) after equating coefficients ofthe terms xq1+1yq2 and xq1yq2+1, we obtain the recurrence formulae

(q1 − q2)u(1)q1,q2 =

q1+q2−1∑s1+s2=0

[(s1 + 1)u(1)s1,s2aq1−s1,q2−s2 − s2u(1)s1,s2bq1−s1,q2−s2 ],

(q1 − q2)u(2)q1,q2 =

q1+q2−1∑s1+s2=0

[s1u(2)s1,s2aq1−s1,q2−s2 − (s2 + 1)u(2)s1,s2bq1−s1,q2−s2 ],

(4.20)

where s1, s2 ≥ −1, q1, q2 ≥ −1, q1+q2 ≥ 0, u(1)1,−1 = u

(1)−1,1 = 0, u

(2)1,−1 = u

(2)−1,1 = 0, u

(1)0,0 = u

(2)0,0 = 1,

and we set aq,m = bm,q = 0 if q + m < 1. It is obvious that u(1)q1,q2 and u

(2)q1,q2 can be computed

by (4.20) when q1 = q2. Denote the polynomials on the right-hand side of (4.20) by ikk and jkk,respectively, when q1 = q2 = k. We see that a sufficient condition for the linearizability of (4.18)is that ikk = jkk = 0 for all k ∈ N. The quantities ikk and jkk are called k-th linearizabilityquantities.

According to Theorem 4.3.2 in [56] the linearizability problem is reduced to calculating thevariety of the ideal L := ⟨i11, j11, i22, j22, . . .⟩ generated by all linearizability quantities. Ageneral approach to completing this task is to find an integer k ≥ 1 such that V(L ) = V(Lk),where Lk denotes the ideal generated by first k pairs of linearizability quantities. Then wecompute the irreducible decomposition of V(Lk) and using appropriate methods show thatall systems from each component of the decomposition are linearizable, that is, the obtainedconditions are sufficient.

One of the most efficient methods to prove linearizability of system (4.18) is the so-calledDarboux linearization, which is a change of variables

z = H1(x, y), w = H2(x, y), (4.21)

where the functions H1 and H2 are of the form

H1(x, y) = f0fα11 · · · fαs

s = x+ . . .

H2(x, y) = g0gβ11 · · · gβt

t = y + . . .

and fi (i = 0, . . . , s), gi (i = 1, . . . , t) are invariant algebraic curves of system (4.18), f0(x, y) =x+ · · · but fj(0, 0) = 1 for 1 ≤ j ≤ s, and g0(x, y) = y + · · · but gj(0, 0) = 1 for 1 ≤ j ≤ t.

Denote by Kj the cofactor of fj and by Lj the cofactor of gj . The transformation H1

linearizes the first equation of (4.18) if

K0 + α1K1 + · · · + αsKs = 1 (4.22)


and the transformation H2 linearizes the second equation of (4.18) if

L0 + β1L1 + · · · + βtLt = −1. (4.23)

If only one of conditions (4.22) and (4.23) is satisfied, let us say (4.23), and system (4.18) has afirst integral ψ(x, y) of the form (2.24), then this system is linearized by the change

z = ψ(x, y)/H2(x, y), w = H2(x, y),

and if (4.23) is satisfied, then the system is linearized by the substitution

z = H1(x, y), w = ψ(x, y)/H1(x, y).

4.3 Bifurcations of critical periods for some families of polynomial systems 75

4.3 Bifurcations of critical periods for some families of polyno-mial systems

In this section we give criteria on the coefficients of a three dimensional system with a centermanifold filled with closed trajectories (corresponding to periodic solutions of the system) todistinguish between the cases of isochronous and non-isochronous oscillations on the centermanifold. Then we study bifurcations of critical periods of these systems.

Since we consider the three-dimensional system only on a center manifold we give in thefirst part of this section some facts about center manifolds. Then, in the second part usingapproaches described in the previous two sections we present our results on isochronicity andcritical periods for a specific family of cubic three-dimensional systems.

4.3.1 Center manifols

In Section 1.1 we gave the Linearization Theorem which allows one to determine the stabilityand qualitative behaviour of the system of differential equations in Rn in a neighborhood of anelementary singular point at the origin.

In this section we present the Local Center Manifold Theorem. It shows that in some casesthe qualitative behaviour in a neighborhood of a singular point x0 of the system

x = f(x) (4.24)

with x ∈ Rn is determined by its behaviour on the center manifold near x0. Since the centermanifold is generally of smaller dimension than system (4.24), this simplifies the problem ofdetermining the stability and qualitative behaviour of the flow near singular point of (4.24). Ofcourse we still must determine the qualitative behaviour of the flow on the center manifold in aneighborhood of the singular point.

Clearly, if f is a C1 function and f(0) = 0 then system (4.24) can be written in the form

w = Jw + G(w), (4.25)

where G(w) is C1 vector function and J is the Jordan form of the matrix A = Df(0) =diag[C,P,Q]. We assume that the square matrix C has c eigenvalues, all with zero real parts,the square matrix P has s eigenvalues, all with negative real parts and the square matrix Q hasu eigenvalues, all with positive real parts, i.e. system can be written in the block diagonal form

x = Cx + F(x,y,z)

y = Py + G(x,y,z)

z = Qz + H(x,y,z),

(4.26)

where (x,y,z) ∈ Rc×Rs×Ru, F(0) = G(0) = H(0) = 0, and DF(0) = DG(0) = DH(0) = 0.

For our purpose we only present the theory for the case when u = 0, that is, when there areno eigenvalues with positive real parts.

Theorem 4.3.1 (The Local Center Manifold Theorem, see e.g. [47]). Let f ∈ Cr(E), where Eis an open subset of Rn containing the origin and r ≥ 1. Suppose that f(0) = 0 and the matrix


of linear part of f has c eigenvalues with zero real parts and s eigenvalues with negative realparts, where c+ s = n. Then system (4.24) can be written in the form

x = Cx + F(x,y)

y = Py + G(x,y),(4.27)

where (x,y) ∈ Rc × Rs, C is a square matrix with c eigenvalues having real parts zero, P isa square matrix with s eigenvalues with negative real parts, and F(0) = G(0) = 0, DF(0) =DG(0) = 0; furthermore, there exists a number δ > 0 and a function h that defines the localcenter manifold

W c(0) = (x,y) ∈ Rc × Rs|y = h(x) for |x|< δ (4.28)

and satisfies

Dh(x)[Cx + F(x,h(x))] − Ph(x) −G(x,h(x)) = 0 (4.29)

for |x|< δ; and the flow on the center manifold W c(0) is defined by the system of differentialequations

x = Cx + F(x,h(x)) (4.30)

for all x ∈ Rc with |x|< δ.

Although equation (4.29) is a quasilinear partial differential equation for the components ofh(x), which can be difficult to solve for h(x), it still gives us a method to compute h(x) to anydegree that we wish, provided that the integer r in Theorem 4.3.1 is sufficiently large. Thisis accomplished by substituting the series expansion for the components of h(x) into equation(4.29). We illustrate this with following example [47].

Example 4.3.2. Consider the family of systems (4.27) with c = 2 and s = 1

x1 = x1y − x1x22

x2 = x2y − x2x21

y = −y + x21 + x22.

(4.31)

We see that in this example we have C =

[0 00 0

], P = [−1] and F(x, y) =

(x1y − x1x

22

x2y − x2x21

)and G(x, y) = x21 + x22.

We substitute the expansion

h(x) = ax21 + bx1x2 + cx22 +O(|x|3)

and

Dh(x) = [2ax1 + bx2, bx1 + 2cx2] +O(|x|2)

into equation (4.29) to obtain

(2ax1 + bx2)[x1(ax21 + bx1x2 + cx22) − x1x

22]

+(bx1 + 2cx2)[x2(ax21 + bx1x2 + cx22) − x2x

21]

+(ax21 + bx1x2 + cx22) − (x21 + x22) +O(|x|3) = 0.


We obtain a = 1, b = 0 and c = 1, thus,

h(x1, x2) = x21 + x22 +O(|x|3).

Substituting this result into equation (4.30) yields

x1 = x31 +O(|x|4)x2 = x32 +O(|x|4)

on the center manifold W c(0) near the origin. The local phase portrait near the origin is givenin Figure 4.1.

Figure 4.1: The phase portrait for the system (4.31).

It should be noted that while there may be many different functions h(x) which determinedifferent center manifolds for (4.26), the flows on the various center manifolds are determinedby (4.30) and they are all topologically equivalent in a neighborhood of the origin.

Note that one of the important applications of center manifolds (called also the Pliss re-duction principle [49]) is that it allows reduction of the study of the stability of the originalhigh-dimensional system to studying the stability of a lower dimensional system.

In the next section we will discuss the case of a 3-dim system of the form

u = −v + P (u, v, w) = P (u, v, w)

v = u+Q(u, v, w) = Q(u, v, s)

w = −λw +R(u, v, w) = R(u, v, w),

(4.32)

where λ is a positive real number and P,Q,R are polynomials without constant or linear terms(if λ is negative, then changing the direction of time and interchanging u and v we still have asystem of the form (4.32) with positive λ). By Theorem 4.3.1 this system has a center manifoldw = f(u, v). There are many systems arising from physics and technology (for instance, theRikitake system [51] for the Earth’s magnetic field or the Hide-Acheson Dynamo [36], and the


Moon-Rand system [46] for control of flexible space structures) which possess a fixed point atwhich the linear part has one negative and two purely imaginary eigenvalues and are, therefore,of the form (4.32).

Since system (4.32) has a center manifold W c and λ > 0, the trajectories in a small neighbor-hood of the origin tend to the trajectories to the center manifold as time increases. In systems(4.32) the phase portrait in a neighborhood of the origin on W c can be, depending on the addednonlinear terms P , Q and R, either a center or a focus. The problem of determining the dynam-ical behavior on W c, that is, distinguishing between a center and a focus on the center manifoldfor a quadratic polynomial system of the form (4.32) was studied in [28].

Since system (4.32) is analytic, for every r ∈ N there exists in a sufficiently small neighbor-hood of the origin a Cr invariant manifold W c, the local center manifold, that is tangent to the(u, v)-plane at the origin and which contains all the recurrent behavior of system (4.32) in aneighborhood of the origin in R3. The following theorem of Lyapunov is proved in [6, §13].

Theorem 4.3.3 (Lyapunov Center Theorem). For system (4.32) with the corresponding vectorfield

X := P ∂∂u + Q ∂

∂v + R ∂∂w

the origin is a center for X|W c if and only if X admits a real analytic local first integral of theform

Φ(u, v, w) = u2 + v2 +∑

j+k+l≥3

ϕjklujvkwl (4.33)

in a neighborhood of the origin in R3. Moreover when there exists a center the local centermanifold W c is unique and is analytic.

As was mentioned above, if we are interested in the behavior of trajectories on the centermanifold of system (4.32) we can find an initial string of the Taylor expansion of the manifoldlooking for it in the form w = a1u+a2v+. . . , then plug the expansion into the first two equationsof system (4.32) and then study the center focus problem for the obtained two-dimensionalsystem. However a computationally more efficient way is provided by the Lyapunov CenterTheorem. By the theorem the existence of a center of X|W c is equivalent to the existence ofa first integral of the form (4.33) for X. So instead of looking for a center manifold we lookfor conditions for existence of an integral Φ of the form (4.33), that is, we look for a functionΦ(u, v, w) in the form (4.33) with undetermined coefficients ϕjkℓ, such that

∂Φ

∂u+∂Φ

∂v+∂Φ

∂w≡ 0. (4.34)

Obstacles to the fulfillment of (4.34) will give us the necessary conditions for the existence of afirst integral of the form (4.33) for system (4.32).

4.3.2 Isochronicity and critical periods of a three-dimensional quadratic sys-tem

In [28] the quadratic system

u = −v + au2 + av2 + cuw + dvwv = u+ bu2 + bv2 + euw + fvww = −w + Su2 + Sv2 + Tuw + Uvw

(4.35)


has been studied and the following results have been derived.

Proposition 4.3.4. A system of the form (4.35) for which S = 0 has a center on the localcenter manifold at the origin.

Theorem 4.3.5. A system of the form (4.35) for which a = b = c + f = 0 and S = 1 has acenter on the local center manifold at the origin if and only if at least one of the following twosets of conditions holds:

1. 8c+ T 2 − U2 = 4(e− d) − T 2 − U2 = 2(e+ d) + TU = 0;2. c = d+ e = 0.

Theorem 4.3.6. A system of the form (4.35) for which d + e = c = f = 0 and S = 1 has acenter on the local center manifold at the origin if and only if at least one of the following threesets of conditions holds:

1. a = b = 0;2. T − 2a = U − 2b = 0.3. d = e = 0.

Since from Proposition 4.3.4 and Theorems 4.3.5 and 4.3.6 we know the conditions for ex-istence of centers on the center manifolds in three subfamilies of [28], the next question thatarises naturally is whether the centers are isochronous. In this section we study isochronicityof the centers and bifurcations of their period functions, that is, the problem of critical periodbifurcations. Since the problems will be reduced to studying two-dimensional systems we willuse the approach for studying isochronicity and critical periods described in Sections 4.1 and4.2.

Now we study isochronicity and critical period bifurcations for subfamilies of system (4.35)listed in Proposition 4.3.4, and Theorems 4.3.5 and 4.3.6.

The case of Proposition 4.3.4

Under conditions of Proposition 4.3.4 system (4.35) is written as

u = −v + au2 + av2 + cuw + dvwv = u+ bu2 + bv2 + euw + fvww = −w + Tuw + Uvw.

(4.36)

It is clear that w = 0 is a center manifold for system (4.36). The corresponding 2D system

u = −v + au2 + av2

v = u+ bu2 + bv2(4.37)

has a center at the origin for all a, b ∈ R (see [28, Theorem 2]).

We now study the isochronicity problem for the above center. Using the computer algebrasystem Mathematica we first turn to the computation of the period function T of system(4.37). After introducing polar coordinates system (4.37) becomes

r = r2(a cosφ+ b sinφ)

φ = 1 + r(b cosφ− a sinφ)(4.38)


and the equation of the trajectories (4.2) is

dr

dφ=

r2(a cosφ+ b sinφ)

1 + r(b cosφ− a sinφ)= R(r, φ). (4.39)

Following the procedure described above we find that w2(φ) = b − b cosφ + a sinφ yieldingT2 = 2π(a2 + b2).

Thus we see that a necessary condition for isochronicity of system (4.37) is a = b = 0, which,obviously, is also a sufficient condition.

To obtain information about critical periods of system (4.37) we investigate the derivativeT ′(r) of period function

T ′(r, (a, b)) = 2T2(a, b)r + 3T3(a, b)r2 + 4T4(a, b)r

3 + · · · . (4.40)

The bifurcations of critical periods of system (4.37) are obtained as the zeros of (4.40) usingthe coefficients T2 = 2π(a2 + b2), T3 = 4πb

(a2 + b2

), T4 = 4π

(a2 + b2

) (2a2 + 3b2

). Using the

results of [56, Section 4.2] we see that T2 = p2. By Proposition 4.2.12 of [56] p2k regarded aspolynomials in variables a and b are homogeneous. The (unit) disc B = (a, b); a2 + b2 ≤ 1can be chosen to be the closed set which is closed under rescaling by nonnegative real constantsand is such that |p2|= |a2 +b2|> 0 for all (a, b) ∈ B\(0, 0), then the result follows by [56, Lemma6.4.2.]. Thus we have the following statement.

Theorem 4.3.7. System (4.36) has an isochronous center at the origin on the center manifoldif and only if a = b = 0 and no critical periods bifurcate from the center of system (4.37).

Case 1 of Theorem 4.3.5

In this case under the same renaming of parameters c, d, e, f, T, U and using α, β as in [28],system (4.35) takes the form

u = −v − 1

2αβuw − 1

2β2vw,

v = u+1

2α2uw +

1

2αβvw,

w = −w + u2 + v2 + (α+ β)uw + (β − α)vw.

(4.41)

A search for invariant algebraic surfaces led to the explicit equation of center manifold W c

given by w = u2+v2

1−αu−βv . Inserting the expression for w into system (4.41) we obtain the system

u = −v − β(αu+ βv)(u2 + v2)

2(1 − αu− βv)

v = u+α(αu+ βv)(u2 + v2)

2(1 − αu− βv).

Using Taylor series expansion up to order five we obtain the system

u = −v +1

2β (uα+ vβ)

(u2 + v2

)+

1

2β (uα+ vβ)2

(u2 + v2

)+m1

v = u− 1

2α (uα+ vβ)

(u2 + v2

)− 1

2α (uα+ vβ)2

(u2 + v2

)+m2,


where

m1 =1

120(−60α3βu5 − 180α2β2u4v + 10u3v2

(−6α3β − 18αβ3

)+ 10u2v3

(−18α2β2 − 6β4

)− 180αβ3uv4 − 60β4v5)

m2 =1

120(60α4u5 + 180α3βu4v + 10u3v2

(6α4 + 18α2β2

)+ 10u2v3

(18α3β + 6αβ3

)+ 180α2β2uv4 + 60αβ3v5).

Further computation following the computational pattern described in Section 4.1 yields againT2 = 1

2π(α2 + β2). This gives the following result.

Theorem 4.3.8. System (4.41) has an isochronous center if and only if α = β = 0 and nocritical periods bifurcate from the center of system (4.41).


For this case we obtain the following system

u = −v + dvw,

v = u− duw,

w = −w + u2 + v2 + (Tu+ Uv)w.

(4.42)

By Theorem 4.3.1 there exists the local center manifold w = F (u, v), where the correspondingtwo dimensional system has the form

u = −v + dvw(u, v),

v = u− duw(u, v).(4.43)

Since in this particular case uu+vv ≡ 0 (for any proper w = F (u, v)), w1 = 1, wk = 0 for k > 1.Computing again the period function we obtain T2 = 2πd.

Thus we see that d = 0 is a necessary condition for isochronicity of the center. Coefficient T2shows again that this is also a sufficient condition. Obviously in this case ⟨T2, T3, T4, . . .⟩ = ⟨T2⟩.Therefore, we have the following result.

Theorem 4.3.9. System (4.42) has an isochronous center if and only if d = 0 and no criticalperiods bifurcate from the center of system (4.42).


The system corresponding to conditions of this case is the same as system with conditions ofCase 2 of Theorem 4.3.5.


A system of this case is of the form

u = −v + au2 + av2 + dvw,

v = u+ bu2 + bv2 − duw,

w = −w + u2 + v2 + 2auw + 2bvw.

(4.44)


System (4.44) has the center manifold w = u2 + v2. Inserting the expression for w intosystem (4.44) yields the system

u = −v + (a+ dv)(u2 + v2)

v = u+ (b− du)(u2 + v2).(4.45)

In order to find all systems with an isochronous center within family (4.45) we first solve theproblem of linearizability for the complexification. To proceed we introduce the new variablex = u+ iv and obtain from (4.45) the complex differential equation

x = ix− idx2x+ (a+ ib)xx.

Adjoining to it its complex conjugate and denoting x by y we obtain the system

x = ix− idx2y + (a+ ib)xy

y = −iy + idxy2 + (a− ib)xy.(4.46)

Introducing new parameters by

a11 = d a01 = −b+ iab11 = d b10 = −b− ia.

(4.47)

system (4.46) is written as

x = i(x− a11x2y − a01xy),

y = −i(y + b11xy2 + b10xy),

(4.48)

where akj , bkj ∈ C. We divide by i and consider akj , bkj as independent parameters (not necessarysatisfying condition (4.47)) and y as an independent unknown function (not necessary satisfyingthe condition y = x) and solve the problem of linearizability for this more general system. Weobtain the following result.

Theorem 4.3.10. System (4.48) is linearizable if and only if one of the following conditionsholds:

1) a01b10 + b11 = b10 = a11 − b11 = 0;

2) a01b10 + b11 = a01 = a11 − b11 = 0.

Proof. We compute first few pairs of linearizability quantities ikk, jkk until a pair is found to liein the radical of the ideal generated by the earlier pairs. After computing the first four pairs wefound that

i22, j22 /∈√

⟨i11, j11⟩

and the third and fourth pair confirm

i33, j33, i44, j44 ∈√

⟨i11, j11, i22, j22⟩,


which leads us to expect that V(L ) = V(L2). Using the Singular routine minAssGTZ wefound that the minimal associated primes of L2 are two ideals Jj written in the statement ofthe theorem. The inclusion V(L ) ⊂ V(L2) is obvious. To prove the reverse inclusion, we haveto show that for every system from Vj for j ∈ 1, 2, there is a transformation z = x + · · ·,w = y + · · · that reduces (4.48) to the linear system z = z, w = −w.

If condition 1) is fulfilled the system has the form

x = x− a01xy,

y = −y.(4.49)

Integrating the system we find also Lyapunov first integral H(x, y) = xye−a01y. Obviously thefirst equation of (4.49) is linearized by the transformation

z = xe−a01y.

Since conditions 2) are dual to conditions 1) under the involution akj ↔ bjk, the system satisfyingthis condition is linearizable as well.

From the last result we can easily determine all systems with an isochronous center infamily (4.48) by substituting equations (4.47) into conditions 1) and 2) of Theorem 4.3.10. Thecomputation gives us the next result regarding to the isochronicity problem for system (4.45).

Corollary 4.3.11. System (4.45) has isochronous center if and only if a = b = d = 0.

We now study critical period bifurcations for system (4.48).Using an obvious modification of the Mathematica code of [56, Fig. 6.4] we have computed

the distinguished normal form of (4.48)

y1 = y1(i+

∞∑j=1

Y(j+1,j)1 (y1y2)

j) y2 = y2(−i+

∞∑j=1

Y(j,j+1)2 (y1y2)

j) (4.50)

and found

Y (2,1) = i(−a11 − a01b10)

Y (1,2) = i(a01b10 + b11)

Y (3,2) = i(−2a01a11b10 − a201b210 − a01b10b11)

Y (2,3) = i(a01a11b10 + a201b210 + 2a01b10b11)

Y (4,3) =i

4(−8a01a

211b10 − 25a201a11b

210 − 9a301b

310 − 20a01a11b10b11 − 14a201b

210b11)

Y (3,4) =i

4(14a201a11b

210 + 9a301b

310 + 20a01a11b10b11 + 25a201b

210b11 + 8a01b10b

211)

Y (5,4) =i

36(−558a201a

211b

210 − 811a301a11b

310 − 235a401b

410 − 360a01a

211b10b11 − 1197a201a11b

210b11

− 525a301b310b11 − 180a01a11b10b

211 − 180a201b

210b

211)

Y (4,5) =i

36(180a201a

211b

210 + 525a301a11b

310 + 235a401b

410 + 180a01a

211b10b11 + 1197a201a11b

210b11

+ 811a301b310b11 + 360a01a11b10b

211 + 558a201b

210b

211).


We compute using (4.17) and obtain

p2 =1

2(a11 + 2a01b10 + b11),

p4 =1

4(a211 + 8a201b

210 + 10a01b10b11 + b211 + 2a11(5a01b10 + b11)),

...

(4.51)

After computing the first four isochronicity quantities we found that√P2 =

√P3 on the

center variety which suggests the following result.

Lemma 4.3.12. For system (4.48) we have

VL = V(Y2) = V(P2) ∩ VC = VI .

Proof. Using Singular, we compute minimal associate primes of Y2 and find that they aregiven by polynomials defining the varieties V1 and V2 of Theorem 4.3.10, which implies thatVL = V(Y2). The second and third equality in the lemma follow by Proposition 4.2.12 andProposition 4.2.7 of [56], respectively.

Using expressions (4.51) we prove that

p2k ∈ ⟨p2, p4⟩ for all k > 2. (4.52)

To this end, we first note that by (4.47) a necessary condition for the existence of a centerof (4.48) is a11 = b11, so we perform this substitution in p2 and p4. Then, using any appropriatecomputer algebra system we first compute a Grobner bases of the ideal P2 = ⟨p2, p4⟩. Com-putations yield that a Grobner basis with respect to the degree-reverse lexicographic order ist1, t2, where t1 = b211, t2 = a01b10 + b11.

Let s = a01b10 and b = b11. It follows from the form of L for (4.48) and Proposition4.2.10 of [56] that for a11 = b11 p2k is a homogeneous polynomial of degree k in s and b.Consider a term smbr of a polynomial p2k (k > 2), (so, m + r = k). If r ≥ 2 then, obviously,smbr ∈ P2. If r = 1 then smb = sm−1b(s + b) − sm−1b2 ∈ P2, and, finally, when r = 0 wehave sm = (s+ b)(sm−1 − sm−2b) + sm−2b2. Since k > 2 we conclude that sm ∈ P2. Therefore,p2k ∈ P2 for all k ∈ N.

To show that there are perturbations which give one critical period, we compute usingexpressions (4.47) first two isochronicity quantities for real system (4.45) and obtain

p2 = a2 + b2 + d

p4 = −2(a2 + b2)2,(4.53)

which confirms again the statement of Corollary 4.3.11. Indeed, the center is isochronous whenthe period function is constant. This is true if and only if all isochronicity quantities are zero.From (4.53) we conclude this occurs if and only if a = b = d = 0. We show that if in system(4.45)

d = −a2 − b2, (4.54)

then one critical period bifurcates from the origin after small perturbations. Inserting (4.53)into T ′(r) we obtain

T ′(r, (a, b, d)) = 2p2(a, b, d)r + 4p4(a, b, d)r3 + · · · . (4.55)


Let system (4.45) with parameters a = a∗, b = b∗, d = d∗ satisfy condition (4.54), that is,d∗ = −a∗2 − b∗2. If a∗2 + b∗2 = 0, then p4 < 0. Choosing d > −a2 − b2 and sufficiently smallwe obtain p2 > 0 and |p2|≪ |p4|, yielding a system with a small root of T ′(r) near the origin. Ifd = a = b = 0 then we first perturb the system in such a way that d = −a2 − b2 and then applythe perturbation described above, again obtaining a critical period of the period function in asmall neighborhood of the origin.

In summary we conclude that the following statement holds.

Theorem 4.3.13. At most one critical period bifurcates from centers on the center manifold ofsystem (4.44) under small perturbations and there are perturbations yielding one critical period.

Case 3. of Theorem 4.3.6

In this case system (4.35) is written as

u = −v + au2 + av2

v = u+ bu2 + bv2

w = −w + u2 + v2 + Tuw + Uvw.(4.56)

The center manifold in this case turns out to be

w =1

2u3(−2a+ 2b+ T − U) + u2v

(−a− b+

T

2+U

2

)+ uv2

(−a+ b+

T

2− U

2

)+ v3

(−a− b+

T

2+U

2

)+ u2 + v2.

However, the corresponding system on the center manifold turns out to be the same as system(4.37) under conditions of Proposition 4.3.4.

Appendix A

To perform rational reconstruction in Mathematica one can use the following code:

RATCONVERT[c_, m_] :=

Block[u = 1,0,m, v = 0,1,c, r,

While[Sqrt[m/2] <= v[[3]], r = u - Quotient[u[[3]], v[[3]]] v; u = v; v = r];

If[Abs[v[[2]]] >= Sqrt[m/2], err, v[[3]]/v[[2]]]]

Given an integer c and a natural number m the function produces a couple of integer numbersv2 and v3 such that v3/v2 ≡ c modm and |v2|, |v3| ≤

√m/2.

87

Appendix B

A Mathematica code for computing focus quantities of system (2.50).

• The operator (2.40) for system (2.50)

l1[nu_1,nu_2,nu_3,nu_4,nu_5,nu_6,nu_7,nu_8]:=

1nu_1+2nu_2+1nu_3+0nu_4+2nu_5+1nu_6+0nu_7+0nu_8



• Definition of function (4.52) of [56].

v[k_1,k_2,k_3,k_4,k_5,k_6,k_7,k_8]:=v[k1,k2,k3,k4,k5,k6,k7,k8]=

Module[us,coef,coef=l1[k1,k2,k3,k4,k5,k6,k7,k8]-l2[k1,k2,k3,k4,k5,k6,k7,k8]; us=0;

v[0,0,0,0,0,0,0,0]=1;

If[k1>0, us=us+(l1[k1-1,k2,k3,k4,k5,k6,k7,k8]+1)*v[k1-1,k2,k3,k4,k5,k6,k7,k8]]

If[k2>0,us=us+(l1[k1,k2-1,k3,k4,k5,k6,k7,k8]+1)*v[k1,k2-1,k3,k4,k5,k6,k7,k8]];

If[k3>0,us=us+(l1[k1,k2,k3-1,k4,k5,k6,k7,k8]+1)*v[k1,k2,k3-1,k4,k5,k6,k7,k8]];

If[k4>0,us=us+(l1[k1,k2,k3,k4-1,k5,k6,k7,k8]+1)*v[k1,k2,k3,k4-1,k5,k6,k7,k8]];

If[k5>0,us=us-(l2[k1,k2,k3,k4,k5-1,k6,k7,k8]+1)*v[k1,k2,k3,k4,k5-1,k6,k7,k8]];

If[k6>0,us=us-(l2[k1,k2,k3,k4,k5,k6-1,k7,k8]+1)*v[k1,k2,k3,k4,k5,k6-1,k7,k8]];

If[k7>0,us=us-(l2[k1,k2,k3,k4,k5,k6,k7-1,k8]+1)*v[k1,k2,k3,k4,k5,k6,k7-1,k8]];

If[k8>0,us=us-(l2[k1,k2,k3,k4,k5,k6,k7,k8-1]+1)*v[k1,k2,k3,k4,k5,k6,k7,k8-1]];

If [coef!=0, us=us/coef]; If [coef==0, gg[k1,k2,k3,k4,k5,k6,k7,k8]=us; us=0]; us]

• gmax is the number of the focus quantities to be computed

gmax=7;

• Computing the quantities q[1],q[2],. . . up to the order ”gmax”

Do[k= sc; num=k; q[num]=0;

For[i1=0,i1<=2 k,i1++ ,

For[i2=0,i2<=(2 k-i1),i2++ ,

For[i3=0,i3<=(2 k-i1-i2),i3++ ,

For[i4=0,i4<=(2 k-i1-i2-i3),i4++,

For[i5=0,i5<=(2 k-i1-i2-i3-i4),i5++,

For[i6=0,i6<=(2 k-i1-i2-i3-i4-i5),i6++,

For[i7=0,i7<=(2 k-i1-i2-i3-i4-i5-i6),i7++,

For[i8=0,i8<=(2 k-i1-i2-i3-i4-i5-i6-i7),i8++,

If[(l1[i1,i2,i3,i4,i5,i6,i7,i8]==k) &&(l2[i1,i2,i3,i4,i5,i6,i7,i8]==k),

v[i1,i2,i3,i4,i5,i6,i7,i8];

q[num]=q[num]+gg[i1,i2,i3,i4,i5,i6,i7,i8]TT[i1,i2,i3,i4,i5,i6,i7,i8]]]]]]]]]],

sc,1,gmax]

89

90 Appendix B

• Definitions of monomials of system (2.50)

TT[l1_,l2_,l3_,l4_,l5_,l6_,l7_,l8_]:=a10^l1 a20^l2 a11^l3 a02^l4 b20^l5 b11^l6 b02^l7 b01^l8

• Output of focus quantities

Do[Print[gg[i]=q[i]], i,1,gmax];

Appendix C

A Mathematica code for computing focus quantities of system (2.62).





v[k_1,k_2,k_3,k_4,k_5,k_6,k_7,k_8]:=v[k1,k2,k3,k4,k5,k6,k7,k8]=

Module[us,coef,coef=l1[k1,k2,k3,k4,k5,k6,k7,k8]-l2[k1,k2,k3,k4,k5,k6,k7,k8]; us=0;

v[0,0,0,0,0,0,0,0]=1;

If[k1>0, us=us+(l1[k1-1,k2,k3,k4,k5,k6,k7,k8]+)*v[k1-1,k2,k3,k4,k5,k6,k7,k8]]

If[k2>0,us=us+(l1[k1,k2-1,k3,k4,k5,k6,k7,k8]+)*v[k1,k2-1,k3,k4,k5,k6,k7,k8]];

If[k3>0,us=us+(l1[k1,k2,k3-1,k4,k5,k6,k7,k8]+)*v[k1,k2,k3-1,k4,k5,k6,k7,k8]];

If[k4>0,us=us+(l1[k1,k2,k3,k4-1,k5,k6,k7,k8]+)*v[k1,k2,k3,k4-1,k5,k6,k7,k8]];

If[k5>0,us=us-(l2[k1,k2,k3,k4,k5-1,k6,k7,k8]+)*v[k1,k2,k3,k4,k5-1,k6,k7,k8]];

If[k6>0,us=us-(l2[k1,k2,k3,k4,k5,k6-1,k7,k8]+)*v[k1,k2,k3,k4,k5,k6-1,k7,k8]];

If[k7>0,us=us-(l2[k1,k2,k3,k4,k5,k6,k7-1,k8]+)*v[k1,k2,k3,k4,k5,k6,k7-1,k8]];

If[k8>0,us=us-(l2[k1,k2,k3,k4,k5,k6,k7,k8-1]+)*v[k1,k2,k3,k4,k5,k6,k7,k8-1]];

If [coef!=0, us=us/coef]; If [coef==0, gg[k1,k2,k3,k4,k5,k6,k7,k8]=us; us=0]; us]

gmax=8;

Do[k= sc; num=k; q[num]=0;

For[i1=0,i1<=2 k,i1++ ,

For[i2=0,i2<=(2 k-i1),i2++ ,

For[i3=0,i3<=(2 k-i1-i2),i3++ ,

For[i4=0,i4<=(2 k-i1-i2-i3),i4++,

For[i5=0,i5<=(2 k-i1-i2-i3-i4),i5++,

For[i6=0,i6<=(2 k-i1-i2-i3-i4-i5),i6++,

For[i7=0,i7<=(2 k-i1-i2-i3-i4-i5-i6),i7++,

For[i8=0,i8<=(2 k-i1-i2-i3-i4-i5-i6-i7),i8++,

If[(l1[i1,i2,i3,i4,i5,i6,i7,i8]==k) &&(l2[i1,i2,i3,i4,i5,i6,i7,i8]==k),

v[i1,i2,i3,i4,i5,i6,i7,i8];

q[num]=q[num]+gg[i1,i2,i3,i4,i5,i6,i7,i8]TT[i1,i2,i3,i4,i5,i6,i7,i8]]]]]]]]]],

sc,1,gmax]

TT[l1_,l2_,l3_,l4_,l5_,l6_,l7_,l8_]:=a30^l1 a21^l2 a12^l3 a03^l4 b30^l5 b21^l6 b12^l7 b03^l8

Do[Print[gg[i]=q[i]], i,1,gmax];

91

Appendix D

The Singular output of primary decomposition of ideal K defined by (3.15).

[1]:

[1]:

_[1]=c10-c12

_[2]=c7*c8-c9^2*c12

_[3]=c6+c8

_[4]=c5+c7

_[5]=c4+c12

_[6]=c2^2-c11

_[7]=c3-c4

_[8]=c1-c2

[2]:

_[1]=c10-c12

_[2]=c7*c8-c9^2*c12

_[3]=c6+c8

_[4]=c5+c7

_[5]=c4+c12

_[6]=c2^2-c11

_[7]=c3-c4

_[8]=c1-c2

[2]:

[1]:

_[1]=c8^2-c9^2*c12

_[2]=c7-c8

_[3]=c6^2-c9^2*c10

_[4]=c5-c6

_[5]=c4*c9^2-c6*c8

_[6]=c4*c8-c6*c12

_[7]=c4*c6-c8*c10

_[8]=c4^2-c10*c12

_[9]=c2^2-c11

_[10]=c3-c4

_[11]=c1-c2

[2]:

_[1]=c8^2-c9^2*c12

_[2]=c7-c8

93

94 Appendix D

_[3]=c6^2-c9^2*c10

_[4]=c5-c6

_[5]=c4*c9^2-c6*c8

_[6]=c4*c8-c6*c12

_[7]=c4*c6-c8*c10

_[8]=c4^2-c10*c12

_[9]=c2^2-c11

_[10]=c3-c4

_[11]=c1-c2

[3]:

[1]:

_[1]=c12

_[2]=c11

_[3]=c10

_[4]=c8

_[5]=c6

_[6]=c4

_[7]=c2

_[8]=c3-c4

_[9]=c1-c2

[2]:

_[1]=c12

_[2]=c11

_[3]=c10

_[4]=c8

_[5]=c6

_[6]=c4

_[7]=c2

_[8]=c3-c4

_[9]=c1-c2

[4]:

[1]:

_[1]=c12

_[2]=c11

_[3]=c10

_[4]=c7

_[5]=c5

_[6]=c4

_[7]=c2

_[8]=c3-c4

_[9]=c1-c2

[2]:

_[1]=c12

_[2]=c11

_[3]=c10

_[4]=c7

95

_[5]=c5

_[6]=c4

_[7]=c2

_[8]=c3-c4

_[9]=c1-c2

[5]:

[1]:

_[1]=c11

_[2]=c10^2-2*c10*c12+c12^2

_[3]=c7*c8-c9^2*c12

_[4]=2*c6*c12+c8*c10+c8*c12

_[5]=2*c6*c10+3*c8*c10-c8*c12

_[6]=2*c6*c7+c9^2*c10+c9^2*c12

_[7]=c6^2+2*c6*c8+c8^2

_[8]=2*c5*c12+c7*c10+c7*c12

_[9]=2*c5*c10+3*c7*c10-c7*c12

_[10]=2*c5*c8+c9^2*c10+c9^2*c12

_[11]=c5*c6-c9^2*c10

_[12]=c5^2+2*c5*c7+c7^2

_[13]=2*c4+c10+c12

_[14]=c2*c10-c2*c12

_[15]=c2*c6+c2*c8

_[16]=c2*c5+c2*c7

_[17]=c2^2

_[18]=c3-c4

_[19]=c1-c2

[2]:

_[1]=c11

_[2]=c10-c12

_[3]=c7*c8-c9^2*c12

_[4]=c6+c8

_[5]=c5+c7

_[6]=c4+c12

_[7]=c2

_[8]=c3-c4

_[9]=c1-c2

[6]:

[1]:

_[1]=c12

_[2]=c11

_[3]=c8

_[4]=c7

_[5]=c5*c6-c9^2*c10

_[6]=c4

_[7]=c2

_[8]=c3-c4

96 Appendix D

_[9]=c1-c2

[2]:

_[1]=c12

_[2]=c11

_[3]=c8

_[4]=c7

_[5]=c5*c6-c9^2*c10

_[6]=c4

_[7]=c2

_[8]=c3-c4

_[9]=c1-c2

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List of Figures

1.1 Phase portraits of singular points of linear systems: (a) saddle, (b) unstable node,(c) unstable focus, (d) center (up to linear transformation). . . . . . . . . . . . . 4

1.2 (a) stable limit cycle, (b) unstable limit cycle, (c)–(d) semistable limit cycles. . . 51.3 A homoclinic orbit Γ, which defines a separatrix cycle. . . . . . . . . . . . . . . . 61.4 Heteroclinic orbits Γ1 and Γ2 defining a separatrix cycle. . . . . . . . . . . . . . . 71.5 Examples of compound separatrix cycles. . . . . . . . . . . . . . . . . . . . . . . 8

4.1 The phase portrait for the system (4.31). . . . . . . . . . . . . . . . . . . . . . . 77

103

Razsirjeni povzetek

V doktorski disertaciji smo se ukvarjali s studijem naslednjih pomembnih problemov teorijenavadnih diferencialnih enacb: s problemom centra in fokusa, s problemom ciklicnosti ter prob-lemoma izohronosti in bifurkacij kriticnih period. Za studij teh problemov smo uporabili metodekomutativne racunske algebre, zato smo v prvem poglavju najprej predstavili nekaj kljucnih poj-mov in algoritmov komutativne racunske algebre.

Glavni predmet nase obravnave so dvodimenzionalni avtonomni polinomski sistemi oblike

x1 = X1(x1, x2), x2 = X2(x1, x2). (1)

Najprej se seznanimo s pojmi, kot so trajektorija, limitni cikli, separatrisne povezave in separa-trisni cikli. Kvalitativno obnasanje druzine (1) je doloceno s tem, kako se x1(t) in x2(t) obnasatas spreminjanjem casa t. Tukaj je primerno obravnavati obnasanje x1(t) in x2(t) v fazni ravnini,t.j. ravnini x1x2, kjer obnasanje predstavimo z druzino krivulj z narascajocim t. Te krivuljese imenujejejo trajektorije ali orbite in geometrijska predstavitev kvalitativne slike trajektorijsistema (1) se imenuje fazni portret. V primeru, ko je sistem linearen (t.j. polinoma na desnistrani obeh enacb v (1) sta linearna), je tip neizrojene singularne tocke v izhodiscu eden izmednaslednjih stirih tipov:

• vozel–stabilni ali nestabilni,

• sedlo,

• fokus (zarisce)–stabilni ali nestabilni,

• center (sredisce).

V primeru, ko je sistem (1) nelinearen (t.j. polinoma X1(x1, x2) in X2(x1, x2) v (1) sta nelin-earna) nam Linearizacijski izrek [4] zagotovi, da imata v okolici elementarne singularne tocke vizhodiscu fazna portreta sistema (1) in njegove linearizacije (t.j. sistem, ki vsebuje samo linearnidel sistema (1)) singularno tocko enakega tipa ob predpostavki, da linearizacija nima centra vizhodiscu. Izolirana zaprta orbita, ki ustreza periodicni resitvi sistema (1), se imenuje limitnicikel. Za karakterizacijo faznega portreta sistema (1) je potrebno najti vse singularne tockesistema in dolociti tip singularnih tock ter najti limitne cikle in trajektorije, ki povezujejo sin-gularne tocke. V doktorskem delu proucujemo predvsem singularne tocke tipa center ali fokuster limitne cikle sistema (1).

V drugem delu prvega poglavja opisemo osnovne pojme teorije polinomskih idealov in glavnelastnosti njihovih raznoterosti. Vpeljemo koncept Grobnerjevih baz in zanesljive tehnike inalgoritme racunske algebre za proucevanje polinomskih idealov in njihovih raznoterosti.

105

106 Razsirjeni povzetek

V drugem poglavju studiramo lokalno integrabilnost dvo-dimenzionalnih analiticnih sistemovdiferencialnih enacb v okolici singularne tocke tipa center ali fokus. Studij tovrstnih sistemovje pomemben del teorije integrabilnosti navadnih diferencialnih enacb. Singularno tocko dvo-dimenzionalnega sistema avtonomnih NDE imenujemo center, ce so vse trajektorije v okolicisingularne tocke zaprte. Ce so vse trajektorije spirale, potem singularno tocko imenujemo fokus.Vsak polinomski sistem s centrom ali fokusom v izhodiscu je lahko zapisan v obliki

du

dt= αu− βv +

∞∑i+j=2

αijuivj ,

dv

dt= βu+ αv +

∞∑i+j=2

βijuivj .

(2)

Center in fokus lahko razlikujemo s pomocjo vpeljave polarnih koordinat v obravnavan sistem(2). S pomocjo sistema (2) v polarnih koordinatah izracunamo Poincarejevo preslikavo

R(r) = η1r + η2r2 + η3r

3 + · · · , (3)

kjer so koeficienti ηj Lyapunova stevila. Poincarejeva preslikava nam pove v katero tocko sevrne trajektorija, ki se zacne na malem odseku Σ = (r, 0), r < r∗ osi u, po enem obratuokrog izhodisca koordinatnega sistema. Ce definiramo funkcijo P(r) := R(r)−r = η1r+η2r

2 +η3r

3 + · · ·, potem je jasno, da ce je P = 0 za vsak 0 < r < r∗ za dovolj majhen r∗, potem so vsetrajektorije v tej okolici izhodisca periodicne in je zato singularna tocka v izhodiscu tipa center.Vidimo, da se to zgodi natanko tedaj, ko so vsa Lyapunova stevila enaka 0. Lyapunova stevilatorej popolnoma dolocajo obnasanje trajektorij v okolici singularne tocke v izhodiscu sistema(2).

Ce v sistemu (2) predpostavimo, da je α = 0, potem je izhodisce fokus. Ce pa predpostavimo,da je α = 0, pa lahko po zamenjavi casa sistem (2) zapisemo v obliki

du

dt= −v +

n∑i+j=2

αijuivj ,

dv

dt= u+

n∑i+j=2

βijuivj . (4)

Poincare in Lyapunov [50, 40] sta pokazala, da je singularna tocka v izhodiscu sistema (4) centernatanko takrat, ko ima sistem (4) prvi integral v obliki

Φ = u2 + v2 +∑

k+l≥2

ϕklukvl. (5)

To je tako imenovan izrek Poincare-Lyapunova. Ta izrek nam pove, da je kvalitativna slikatrajektorij v okolici singularne tocke povezana z lokalno integrabilnostjo sistema: singularnatocka je center natanko takrat, ko obstaja analiticni prvi integral. Se vec, Poincare in Lyapunovsta pokazala, da analiticni prvi integral sistema (4) obstaja natanko takrat, ko obstaja formalniprvi integral sistema (4) v obliki (5). Vendar metoda Poincareja in Lyapunova ne da odgovorana vprasanje kako ugotovimo ali za sistem diferencialnih enacb obstaja prvi integral v obliki(5). Odgovor na to vprasanje je potrebno najti za vsak sistem posebej, zaenkrat nimamo uni-verzalnih metod, ki bi omogocile zanesljiv odgovor za splosen sistem (4). Torej obstaja problemkako prepoznamo sisteme s centrom znotraj dane parametricne druzine ravninskih polinomskih

107

sistemov navadnih diferencialnih enacb. Ta problem je prvi obravnaval Dulac leta 1908 [26], kiga je resil za primer kvadraticnega sistema.

Ceprav je o problemu centra in fokusa veliko napisanega, je le-ta resen le za kubicne sisteme vobliki linearnega centra motenega s homogenimi kubicnimi nelinearnostmi [45], za tako imenovanKuklesov sistem [43, 63], za nekatere linearne centre perturbirane s homogenimi polinomi petestopnje [13] in za nekaj posebnih druzin polinomskih sistemov NDE. Celo primer kubicnegasistema, to je sistem v obliki (4) z n = 3 je se zmeraj neresen. Ena izmed glavnih tezav v studijuproblema centra izhaja iz izracunov raznoterosti ideala generiranega z Lyapunovimi kolicinami,ki so koeficienti Poncarejeve preslikave. V bistvu je to algebraicen problem: izracunati mnozicoskupnih nicel polinomskega ideala in najti ireducibilno dekompozicijo te mnozice. Kakorkoli,tezava je posledica dejstva, da ne poznamo generirajoce mnozice ideala ampak samo rekurzivnoformulo za izracun polinomov ideala.

Nas pristop temelji na kompleksifikaciji realnega sistema (4), saj na ta nacin postanejopolinomi elementi kolobarja C[x1, . . . , xn] in izracun raznoterosti idealov, generiranih s polinomi,odvisnimi od koeficientov sistema, s katerimi se ukvarjamo pri proucevanju problema lokalneintegrabilnosti, postane enostavnejsi. Ce vpeljemo novo spremenljivko x = u + iv, dobimo izsistema (4) enacbo

x = P (x, x) = i(x−n−1∑

p+q=1

apqxp+1xq), (6)

ki ji recemo polinomski sistem v kompleksni obliki. Ce tej enacbi dodamo konjugirane koeficientebqp = apq in recemo, da je y = x, potem dobimo sistem diferencialnih enacb

x = i(x−n−1∑

p+q=1

ap,qxp+1yq) = P (x, y)

y = −i(y −n−1∑

p+q=1

bq,pxqyp+1) = Q(x, y).

(7)

Po spremembi casa idt = dτ in zamenjavi t namesto τ sistem (7) postane

x = x−n−1∑

p+q=1

ap,qxp+1yq = x− P (x, y)

y = −y +

n−1∑p+q=1

bq,pxqyp+1 = −y +Q(x, y),

(8)

kjer so vsi koeficienti apq, bqp ∈ C in obe spremenljivki x, y ∈ C. Sistem (7) (in tudi (8)) seimenuje kompleksifikacija sistema (4).

Ce predpostavimo v (7), da y = x in bqp = apq, potem ima sistem (7) realno prasliko.V tej doktorski nalogi smo pri obravnavi problema centra za posamezne polinomske druzinepredpostavili, da je sistem (7) splosen brez zahteve, da je bqp = apq. Na ta nacin dobimo zeljenoinformacijo oz. pogoje za center za kompleksen dvo-dimenzionalen sistem. Ob prehodu nazaj narealen sistem (4) dobimo pogoj za center realnega sistema (4). Sistema (7) in (8) imata centerv izhodiscu natanko tedaj, ko obstaja analiticni prvi integral oblike

Ψ(x, y) = xy +∑

j+k≥3

vj−1,k−1xjyk. (9)


Resiti problem centra za kompleksni sistem (7) (ali (8)) pomeni najti potrebne in zadostnepogoje za obstoj prvega integrala oblike (9). V splosnem ta integral ne obstaja vedno. Ceposkusimo najti integral, pridemo do vrste oblike (9), za katero je izraz

DΨ :=∂Ψ

∂xP +

∂Ψ

∂yQ

enak

DΨ = g11(xy)2 + g22(xy)3 + g33(xy)4 + · · · . (10)

Iz definicije prvega integrala vemo, da mora veljati DΨ ≡ 0, ce zelimo, da je Ψ prvi integral.Odtod pa sledi, da morajo vsi polinomi gkk v (10) biti 0. Na ta nacin dobimo pogoje za obstojprvega integrala (9) za sistem (7) oz. pogoje za obstoj centra v izhodiscu sistema (7).

Za lazje razumevanje naslednje definicije, se spomnimo, da je za ideal I, generiran s polinomif1, . . . , fs ∈ k[x1, . . . , xn] (k je polje), njegova raznoterost

V(I) = a = (a1, . . . , an) ∈ kn : fj(a) = 0 za 1 ≤ j ≤ s

in njegov radikal

√I = f ∈ k[x1, . . . , xn] : obstaja p ∈ N, da je fp ∈ I.

Definicija 1. Naj bo C[a, b] kolobar polinomov v spremenljivkah parametrov ap,q in bq,p sistema(7) s koeficienti v C. Polinom gkk ∈ C[a, b] na desni strani (10) se imenuje k-ta fokusnakolicina za singularnost v izhodiscu sistema (7) (ali (8)). Ideal, generiran s fokusnimi kolicinami,B = ⟨g11, g22, . . . , gjj , . . .⟩ ⊂ C[a, b], se imenuje Bautinov ideal, in afina raznoterost VC = V(B)se imenuje raznoterost centra za singularnost v izhodiscu sistema (7) (ali (8)).

Z Bk oznacimo ideal generiran s prvimi k fokusnimi kolicinami, Bk = ⟨g11, g22, . . . , gkk⟩.Torej je iskanje pogojev za center v izhodiscu sistema (7) ekvivalentno iskanju raznoterosti

centra VC . To naredimo tako, da z uporabo sistema racunske algebre (mi smo uporabili sistemMathematica) izracunamo prvih nekaj fokusnih kolicin in nato najdemo taksen najmanjsi K,da veriga radikalov

√B1 ⊂

√B2 · · · stabilizira, kar pomeni, da

√B1 ⊂ · · · ⊂

√BK =

√BK+1.

Ko enkrat najdemo tak K, potem z uporabo sistema racunske algebre Singular izracunamoireducibilno dekompozicijo raznoterosti V(BK) in na ta nacin dobimo domnevne komponenteraznoterosti centra. Za vsako od teh komponent potem z uporabo ustreznih metod poiscemo prviintegral oblike (9). Eden izmed najucinkovitejsih orodij za iskanje prvih integralov je Darbouxovametoda. Poleg te je v nalogi uporabljena tudi casovna reverzibilnost sistema, preverjanje, ce jesistem Hamiltonski, pa se nekatere druge metode.

Na ta nacin smo v doktorski disertaciji resili problem integrabilnosti za naslednji dve druzinidvo-dimenzionalnih polinomskih sistemov:

x = x(1 − a10x− a20x2 − a11xy − a02y

2),

y = −y(1 − b01y − b02y2 − b11xy − b20x

2),(11)

x = x(1 − a30x3 − a21x

2y − a12xy2 − a03y

3),y = −y(1 − b30x

3 − b21x2y − b12xy

2 − b03y3),

(12)

109

in deloma za sistem

x = x− a40x5 − a31x

4y − a22x3y2 − a13x

2y3 − a04xy4 − y5,

y = −y + x5 + b40x4y + b31x

3y2 + b22x2y3 + b13xy

4 + b04y5.

(13)

Za sistem (11) imamo naslednji rezultat.

Izrek 2. Naj bo V(B) raznoterost Bautinovega ideala druzine (11) in naj bo B4 = ⟨g11, g22, g33, g44⟩.Potem je V(B) = V(B4). Se vec, V(B) sestoji iz naslednjih petih ireducibilnih komponent

V(B) = V(J1) ∪V(J2) ∪V(J3) ∪V(J4) ∪V(J5),

kjerJ1 = ⟨a210b02 − a20b

201, a02a20 − b20b02, a02a

210 − b20b

201, a11 − b11⟩

J2 = ⟨a20 + b20, a11 − b11, a02 + b02⟩J3 = ⟨a11, a02, b02, b11⟩J4 = ⟨a20, a11, b11, b20⟩J5 = ⟨a11, a02, b11, b20⟩.

Za sistem (12) se je problem centra izkazal za veliko tezjega, kot je bil ta problem za prejsnjisistem (11). Zaradi teze izracunov smo sistem razdelili na tri poddruzine. Opazimo, da ce jev sistemu (12) a12 = 0 in b21 = 0, potem ga lahko z linearno transformacijo spremenimo vekvivalentni sistem, kjer je a12 = b21 = 1. Tako lahko sistem (12) razdelimo na tri primere:

(α) a12 = b21 = 1, (β) a12 = 1, b21 = 0, (γ) a12 = b21 = 0.

V primerih (α) in (β) se je izkazalo, da sta primera se vedno racunsko zahtevna, saj je bilonemogoce izracunati ireducibilno dekompozicijo nad karakteristiko 0. Zato smo se lotili izracunavnad poljem karakteristike 32003 (t.j. poljem Z32003). Tako smo uporabili dekompozicijski algo-ritem z modularnimi aritmetikami [55], ki je podan v naslednjih petih tockah.

• Izberemo prastevilo p in izracunamo minimalne prikljucene praideale Q1, . . . , Qs ideala Iv Zp[x1, . . . , xn].

• Z uporabo algoritma za racionalno rekonstrukcijo spremenimo ideale Qi (i = 1, . . . , s)v ideale Qi ⊂ Q[x1, . . . , xn] (t.j. zamenjamo vse koeficiente v Qi z racionalnimi stevili,izracunanimi z rekonstrukcijskim algoritmom [67]).

• Za vsak i = 1, . . . , s z uporabo testa za clanstvo v idealu preverimo, ce so polinomif1, . . . , fs v radikalu ideala Qi, t.j. ce je reducirana Grobnerjeva baza ideala ⟨1 − wf,Qi⟩enaka 1 za vsak i = 1, . . . , s. Ce je odgovor ”da”, potem gremo na korak 4, sicer izberemodrugo prastevilo in gremo nazaj na korak 1.

• Izracunamo Q = ∩si=1Qi ⊂ Q[x1, . . . , xn].

• Preverimo√Q =

√I, t.j. za vsak g ∈ Q je reducirana Grobnerjeva baza ideala ⟨1−wg, I⟩

enaka 1 in za vsak f ∈ I je reducirana Grobnerjeva baza ideala ⟨1 − wf,Q⟩ enaka 1.Ce to drzi, potem je V(I) = ∪s

i=1V(Qi), sicer izberemo drugo prastevilo p in se vrnemona korak 1.


Za primer (α) smo pridobili potrebne pogoje za kompleksni center v izhodiscu sistema (12),ki so predstavljeni v izreku 3. Prav tako smo dokazali zadostnost teh pogojev, razen za primer10). Preostala primera sta obravnavana v izreku 4 in izreku 5.

Izrek 3. Naslednji pogoji so potrebni pogoji za kompleksni center v izhodiscu sistema (12) za12 = b21 = 1:

1) a21−b12 = 2a03b12+b12b03−a03−2b03 = a30b12+2b30b12−2a30−b30 = a30a03−b30b03 = 0;

2) a30 − b30 = a21 − b12 = a30 − b03 = 0;

3) b03 = b12 − 2 = a03 = a21 − 2 = 0;

4) b12 − 2 = b30 = a21 − 2 = a30 = 0;

5) b12 − 2 = a03 − 2b03 = a21 − 2 = 2a30 − b30 = 0;

6) b03 − 1 = b30 + b12 − 2 = a03 − b12 + 2 = a21 + b12 − 4 = a30 − 1 = 0;

7) 12a03 − 8b12 − 3b03 + 7 = 3a30 + 8a21 − 12b30 − 7 = 2b30b03 − a21 + b12 = 0,8a21b03 + 12b12 − 7b03 + 3 = 32b212 − 20b12 + 27b03 − 7 = 8b30b12 + 3a21 − 7b30 − 6 = 0,8a21b12 − 7a21 − 7b12 − 4 = 8a221 − 23a21 + 27b30 + 14 = 0;

8) b03 − 4 = b12 + 9 = 4b30 − 5 = a03 − 8 = a21 + 2 = 4a30 + 1 = 0;

9) 4b03 + 1 = b12 + 2 = b30 − 8 = 4a03 − 5 = a21 + 9 = a30 − 4 = 0;

10) 9b03 + 16 = 3b12 + 5 = 16b30 + 9 = 9a03 + 8 = 2a21 + 1 = 16a30 − 3 = 0;

11) 4b03 − 1 = b12 − 1 = b30 + 4 = a03 = a21 = a30 + 8 = 0;

12) 16b03 − 3 = 2b12 + 1 = 9b30 + 8 = 16a03 + 9 = 3a21 + 5 = 9a30 + 16 = 0;

13) b03 + 8 = b12 = b30 = a03 + 4 = a21 − 1 = 4a30 − 1 = 0.

Ce velja eden od pogojev 1)-9), 11)–13), potem ima ustrezen sistem center v izhodiscu.

Izrek 4. Sistem (12) z a12 = 1, b21 = 0 ima kompleksni center v izhodiscu natanko tedaj, kovelja eden izmed naslednjih pogojev:

1) a21 = 2a03b12 + b12b03 − a03 − 2b03 = a30b12 + 2b30b12 − 2a30 − b30 = a30a03 − b30b03 = 0;

2) b03 = b12 − 2 = a03 = a21 = 0;

3) b30 = a21 = a30 = 0;

4) b12 − 2 = a03 − 2b03 = a21 = 2a30 − b30 = 0;

5) 8b12 − 7 = 4a03 + b03 = a30 − 4b30 = 2b30b03 − a21 = 16a21b03 + 27 = 8a221 + 27b30 = 0;

6) b12 = b30 = 2a03 + b03 = a30b03 − 2a21 = a21b03 + 8 = a221 + 4a30 = 0;

7) b12 = b30 = 2a03 − b03 = a30b03 + 2a21 = a21b03 − 8 = a221 + 4a30 = 0.

111

Izrek 5. Sistem (12) z a12 = b21 = 0 ima kompleksni center v izhodiscu natanko tedaj, ko veljaeden izmed naslednjih pogojev:

1) b03 = a03 = a21 = 0;

2) a30a03 − b30b03 = a30b312 − a321b03 = a321a03 − b30b

312 = 0;

3) b12 = b30 = a30 = 0;

4) 2a03 + b03 = a30 + 2b30 = 0;

5) a03 − 2b03 = 2a30 − b30 = 2a21b12 − 9b30b03 = 0.

V studiji problema centra za sistem (13) smo zaradi zahtevnosti izvedli izracunave s pomocjomodularne aritmetike. Iz istega razloga nismo obravnavali celega sistema, ampak naslednjepoddruzine:

(C1) a40 = b04 = 0, (C2) a31 = b13 = 0, (C3) a13 = b31 = 0, (C4) a04 = b40 = 0.

Za vecino dobljenih potrebnih pogojev smo dokazali, da so tudi zadostni pogoji za obstojcentra v izhodiscu ustreznih sistemov.

Naslednji problem, ki je obravnavan v doktorski nalogi je problem ciklicnosti dvo-dimenzionalnegapolinomskega sistema navadnih diferencialnih enacb. Ta problem je tesno povezan s 16. Hilber-tovim problemom o stevilu izoliranih periodicnih resitev (limitnih ciklov) za dvo-dimenzionalnepolinomske sisteme

x = Pn(x, y), y = Qn(x, y) (14)

(n je stopnja polinomov na desni strani). Navkljub dejstvu, da je bil 16. Hilbertov problemformuliran vec kot sto let nazaj, se ni resen niti za kvadraticni sistem, to je za sistem (14) zn = 2.

Bistven del tega Hilbertovega problema je ocena maksimalnega stevila limitnih ciklov, kilahko bifurcirajo iz singularne tocke tipa center ali fokus pod vplivom majhnih motenj koefi-cientov sistema, t.i. problem ciklicnosti. Za podrobnejsi opis tega problema podamo naslednjodefinicijo [56].

Definicija 6. Naj bo E podmnozica od Rn in F : R × E → R : (z, θ) 7→ F (z, θ) analiticnafunkcija, ki jo v okolici z = 0 zapisemo kot

F (z, θ) =

∞∑j=0

fj(θ)zj , (15)

kjer so fj(θ) analiticne funkcije za j ∈ N0 in vrsta (15) je konvergentna v okolici (z, θ) za vsakθ∗ ∈ E . Ce je θ∗ ∈ E taksna, da je F (z, θ∗) = 0 in nz,ε oznacuje stevilo izoliranih nicel funkcijeF (z, θ) na intervalu 0 < z < ε, potem recemo, da je multiplikativnost funkcije F (z, θ) v θ∗

glede na mnozico E enaka m, ce obstajata taksna δ0 > 0 in ε0 > 0, da za vsak par (ε, δ), kizadosca pogoju, da je 0 < ε < ε0 in 0 < δ < δ0, velja

maxθ∈Uδ(θ∗)∩E

nθ,ε = m.


Problem ciklicnosti je problem multiplikativnosti funkcije P(r) = R(r) − r in nθ,ε v temprimeru ustreza stevilu limitnih ciklov v majhni okolici izhodisca sistema (4).

Z uporabo algoritmov racunske algebre lahko v primeru, ko je Bautinov ideal radikalen,problem resimo na relativno enostaven nacin [52]. Kakorkoli, situacija, ko je Bautionov idealradikalen, je zelo redka in do sedaj se ni ucinkovitih metod za studij problema ciklicnosti zagenericne polinomske sisteme. Pred kratkim je bil predlagan pristop, ki je uspesen pri nekaterihsistemih z neradikalnim Bautinovim idealom [39]. Ucinkovito metodo, katera deluje v mnogihprimerih, je predlagal tudi C. Christopher [20].

V doktorski nalogi je resen problem ciklicnosti za realni kubicni sistem (14) zapisan v kom-pleksni obliki

x = λx+ ix(1 − a10x− a20x2 − a11xx− a02x

2), (16)

kjer je x = u+ iv. Dobili smo naslednji rezultat.

Izrek 7. Ciklicnost centra v izhodiscu sistema (16), za katerega velja |a20|+|a11|+|a02|= 0, jekvecjemu stiri.

V dokazu tega rezultata smo najprej preverili, ce je Bautinov ideal radikalen. Ker je odgovornegativen, smo uvedli nove spremenljivke, povezane z idealom Sibirskega, ki opisuje mnozicocasovno reverzibilnih sistemov. Z analizo le-teh smo potem dobili informacijo o stevilu za zgornjomejo ciklicnosti.

S pomocjo studija vsake posamezne komponente raznoterosti centra za realni sistem smodokazali, da obstajajo sistemi, kjer je ciklicnost natanko stiri. Ker je ciklicnost vsake komponenteraznoterosti centra povezana z njeno dimenzijo, smo izracunali dimenzijo vsake komponente innasli natancno ciklicnost genericnih sistemov iz komponent.

Ce se za nek sistem da ugotoviti, da ima center, se pojavi naslednje vprasanje, ali je centerizohron. Po definiciji, ce imajo vse periodicne resitve v okolici centra isto periodo, tedaj jecenter izohron. Problem izohronosti je najti pogoje, pri katerih bo center izohron. Izohronostso raziskovali ze vsaj v 17. stoletju, ko je Chrisitian Huygens opazil, da je perioda nihanja uremonotona funkcija energije in se zmanjsuje, ko se zmanjsuje energija, t.j. ko se urina kroznavzmet odvija. Problem je ponovno pritegnil zanimanje v zgodnjih sestdesetih letih prejsnjegastoletja, ko je japonski matematik Minoru Urabe raziskoval izohronost nekaterih hamiltonskihsistemov [66].

Se en problem, tesno povezan s problemi centra, izohronosti in ciklicnosti, je problem bi-furkacij kriticnih period, torej bifurkacij funkcije periode sistema. Problem bifurkacij kriticnihperiod sta proucevala C. Chicone in M. Jacobs [18]. Ce v obravnavan sistem (4) vpeljemopolarne koordinate, potem lahko izracunamo t.i. funkcijo periode v obliki

T (r) = 2π

(1 +

∞∑k=1

Tkrk

), (17)

kjer T (r) poda cas, ki je potreben, da se trajektorija, ki se zacne za t = 0 v tocki (r, 0) majhnegaodseka (r∗, 0) osi u, prvic vrne nazaj na ta odsek. Iz (17) vidimo, da bo funkcija periode (17)konstantna (to pomeni, da bo sistem izohron), ko bodo vsi koefienti Tk enaki 0. V [18] sta avtorjapokazala, da je lahko problem bifurkacij kriticnih period reduciran na problem iskanja bazeideala ⟨T2, T4, T6, . . .⟩. Problem bifurkacij kriticnih period, ki je definiran kot maksimalno stevilo

113

kriticnih period, ki bifurcirajo iz singularne tocke tipa center, pa je problem multiplikativnosti(v smislu definicije 6) odvoda funkcije periode T ′(r).

Spomnimo se sedaj pojma centralne mnogoterosti. Obravnavamo sistem

x = Cx + F(x,y)

y = Py + G(x,y),(18)

kjer (x,y) ∈ Rc×Rs, C je kvadratna matrika s c lastnimi vrednostmi, ki imajo nicelni realni del,P je kvadratna matrika s c lastnimi vrednostmi, ki imajo negativni realni del in F, G ∈ Cr(E),kjer je E odprta podmnozica Rn, n = c+ s, F(0) = G(0) = 0, DF(0) = DG(0) = 0. Nadaljeobstaja δ > 0 in funkcija h, ki definira lokalno centralno mnogoterost

W c(0) = (x,y) ∈ Rc × Rs|y = h(x) for |x|< δ (19)

in zadosca pogoju

Dh(x)[Cx + F(x,h(x))] − Ph(x) −G(x,h(x)) = 0 (20)

za |x|< δ; in tok na centralni mnogoterosti W c(0) je definiran s sistemom diferencialnih enacb

x = Cx + F(x,h(x)) (21)

za vse x ∈ Rc z|x|< δ. Iz zgoraj navedenega, ki v bistvu predstavlja izrek o lokalni centralnimnogoterosti (glej npr. [47]) vidimo, da je zagotovljen obstoj funkcije h(x), ki zadosca pogoju(20).

V doktorski disertaciji smo obravnavali problema izohronosti in bifurkacij kriticnih periodtri-dimenzionalnega sistema, ki je bil proucen v [28] z vidika obstoja prvih integralov in central-nih mnogoterosti. Resena sta problema izohronosti in bifurkacij kriticnih period na centralnimnogoterosti za tri-dimenzionalni realni kvadraticni sistem oblike

u = −v + au2 + av2 + cuw + dvwv = u+ bu2 + bv2 + euw + fvww = −w + Su2 + Sv2 + Tuw + Uvw.

(22)

V [28] so bili dobljeni naslednji pogoji za obstoj centra na centralni mnogoterosti:

(a) S = 0;

(b) a = b = c+ f = 8c+ T 2 − U2 = 4(e− d) − T 2U2 = 2(e+ d) + TU = 0 and S = 1;

(c) a = b = c = f = d+ e = 0 in S = 1;

(d) d+ e = c = f = T − 2a = U − 2b = 0 in S = 1;

(e) c = d = e = f = 0 in S = 1.

Tako smo obravnavali sistem (22), ki zadosca enemu izmed petih pogojev zgoraj.Ce velja pogoj (a), potem sistem (22) postane

u = −v + au2 + av2 + cuw + dvwv = u+ bu2 + bv2 + euw + fvww = −w + Tuw + Uvw.

(23)

Centralna mnogoterost za ta sistem je w = 0. Za sistem (23) smo pridobili naslednji rezultat.


Izrek 8. Sistem (23) ima izohroni center v izhodiscu na centralni mnogoterosti w = 0 natankotedaj, ko je a = b = 0 in nobena kriticna perioda ne bifurcira iz centra sistema (23).

Ce vstavimo pogoj (b) v sistem (22) in ustrezno preimenujemo koeficiente c, d, e, f, T , U teruporabimo koeficienta α, β, potem dobimo sistem

u = −v − 1

2αβuw − 1

2β2vw,

v = u+1

2α2uw +

1

2αβvw,

w = −w + u2 + v2 + (α+ β)uw + (β − α)vw,

(24)

ki ima centralno mnogoterost w = u2+v2

1−αu−βv . Izkaze se, da za ta sistem velja naslednji izrek.

Izrek 9. Sistem (24) ima izohron center natanko tedaj, ko α = β = 0 in nobena kriticna periodane bifurcira iz centra sistema (24) na centralni mnogoterosti

Za primer (c) dobimo sistem

u = −v + dvw,

v = u− duw,

w = −w + u2 + v2 + (Tu+ Uv)w.

(25)

Ta sistem ima lokalno centralno mnogoterost w = F (u, v), kjer ima ustrezen dvo-dimenzionalensistem obliko

u = −v + dvw(u, v),

v = u− duw(u, v).(26)

Ce izracunamo funkcijo periode, dobimo naslednji rezultat.

Izrek 10. Sistem (25)ima izohroni center na centralni mnogoterosti natanko tedaj, ko je d = 0in nobena kriticna perioda ne bifurcira iz centra sistema (25).

Sistem (22) s pogojem iz (d) postane

u = −v + au2 + av2 + dvw,

v = u+ bu2 + bv2 − duw,

w = −w + u2 + v2 + 2auw + 2bvw.

(27)

Sistem (27) ima centralno mnogoterost w = u2 + v2.

Izrek 11. Najvec ena kriticna perioda bifurcira iz centra na centralni mnogoterosti sistema (27)in obstajajo motnje, ki ustvarijo eno kriticno periodo.

Nazadnje se obravnavamo primer (e), vendar smo za ta primer ugotovili, da je ekvivalentenprimeru (a).

Professional Curriculum Vitae

Personal informationAddress: CAMTP-Center for Applied Mathematics and Theoretical PhysicsUniversity of Maribor, Krekova 2, SI-2000 Maribor, Slovenia.e-mail: [email protected] of Birth: May 25, 1985Place of Birth: Ptuj, SloveniaCitizenship: SlovenianGender: Female

Formal Education

2009-PhD in MathematicsThesis Topic:Integrability and local bifurcations in polynomial systems of ordinary differentialequationsMentor: Dr. Valery RomanovskiCo-mentor: Professor Dr. Douglas ShaferArea of Study: Differential equations, Dynamical systems.

2004-2009Graduation: Teacher of MathematicsMentor: Professor Dr. Matej Bresar.

2000 - 2004High school in Ptuj.

1991 - 2000Primary school in Cirkulane.

Scientific Papers - Published

1. B. Fercec, X. Chen and V. G. Romanovski. Integrability conditions for complex systemswith homogeneous quintic nonlinearities. Journal of Applied Analysis and Computation(2011), no. 1, 9–20.

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116 Professional Curriculum Vitae

2. B. Fercec, J. Gine, Y. Liu, and V. G. Romanovski. Integrability conditions for Lotka-Volterra planar complex quartic systems having homogeneous nonlinearities. Acta appl.math. 124 (2013) 107–124.

3. B. Fercec, M. Mencinger, and V. G. Romanovski. Investigation of center manifolds ofthree-dimensional systems using computer algebra. Programirovanie no. 2 (2013) 21–30(in Russian); Programing and Computer Software 39 (2013) 67–73 (English translation).

4. B. Fercec and A. Mahdi. Center conditions and cyclicity for a family of cubic systems:Computer algebra approach. Math. Comput. Simul. 87 (2013) 55–67.

Scientific Papers (Conference Proceedings)

1. B. Fercec, X. Chen and V. G. Romanovski. Integrability conditions for complex systemswith homogeneous quintic nonlinearities (to appear in the Proceedings of the VIII Inter-national School/Conference on Group Theory, Nalchik, Russia, 5-10 July 2010).

2. B. Fercec and M. Mencinger. Isochronicity of centers at a center manifold. In: ROBNIK,Marko, ROMANOVSKI, Valery. ”Let’s Face Chaos through Nonlinear Dynamics”, (AIPconference proceedings, 1468). Melville, N.Y.: American Institute of Physics, (2012) 148–157.

Conferences,Workshops and Meeting

1. 12th Japan-Slovenia Seminar on Nonlinear Science, October 7 - 9, 2009, Maribor, Slovenia.

2. 8th Christmas Symposium of Physicists, December 17 - 19, 2009, Maribor, Slovenia.

3. Symbolic Computation and its Applications, June 30 - July 2, 2010, Maribor, Slovenia.

4. VIII International School/Conference on Group Theory, July 5 - 10, 2010, Nalchik, Russia.

5. The 18th Conference on Applied and Industrial Mathematics: Caim 2010, October 14 -17, 2010, Iasi, Romania.


7. The Fifth International Conference on Recent Advances in Applied Dynamical Systems,May 16 - 18, 2011, Shanghai, China.

8. 8th International Summer School/Conference: ”Let’s Face Chaos Through Nonlinear Dy-namics” , June 26 - July 10, 2011, Maribor, Slovenia.


10. The 14th International Workshop on Computer Algebra in Scientific Computing, Septem-ber 3 - 6, 2012, Maribor, Slovenia


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Visits

1. Department of Mathematics, Sichuan University in Chengdu, China, January 14 - 24,2010. I gave a talk with the title ”Integrability of some quintic systems.”

2. Shanghai Normal University, Shanghai, China, May 12 - 21, 2011. At the seminar ofthe Department of Mathematics I gave a talk with the title ”Integrability conditions forcomplex systems with homogeneous nonlinearities.”

3. Department of Mathematics and Statistics, University of North Carolina at Charlotte,USA, July 1 - October 28, 2011. We investigated the cyclicity of the subfamily of cubicsystems and we obtained the conditions for a center and an upper bound for the numberof limit cycles of this subfamily of cubic systems. We also computed the cyclicity of eachirreducible component of center variety. Then we started to study bifurcations of criticalperiods for some families of polynomial systems of ODE’s.

4. Department of Mathematics, Sichuan University in Chengdu, China, December 14 - 21,2011. We studied Z2-reversible-equivariant planar systems.

INTEGRABILITY AND LOCAL BIFURCATIONS IN POLYNOMIAL … · cikliˇcnosti, bifurkacije kritiˇcnih...

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