On the Center Problem for Ordinary Differential Equations

UNIVERSITY OF CALGARY

Douglas McLean

A THESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF Masters of Science

DEPARTMENT OF MATHEMATICS AND STATISTICS

CALGARY, ALBERTA August, 2010

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1+1 Canada Table of Contents

1 Introduction 1 1.1 Henri Poincare 2 1.2 David Hilbert 5 1.3 Jean Gaston Darboux 7 1.4 Henri Dulac 8 1.5 Yulij S. Ilyashenko 8 1.6 Nikolai Nikolaevich Bautin 9

2 Classical Results 10 2.1 Bautin's Theorem 10 2.2 The Center Conditions for Degree 2 Planar Polynomial Systems ... 24 2.3 Cherkas Transform 26 2.4 Abel Equations Obtained from Vector Fields of Degree 2 29

3 Algebraic Methods 34 3.1 Introduction 34 3.2 An Explicit Expression for the First Return Map 34 3.3 The Universal Center 42 3.4 Systems Generating the Group of Rectangular Paths 54 3.4.1 Group of Piecewise Linear Paths 58 3.4.2 Center Problem for the Group of Rectangular Paths 59

3.4.3 Bautin Problem for the group of rectangular paths 64

4 Further Work 68

Bibliography 69

n List of Figures

1.1 Henri Poincare 3 1.2 David Hilbert 5 1.3 Jean Gaston Darboux 7 1.4 Yulij S. Ilyashenko 9

2.1 A Vector Field with a Center 11 2.2 A Limit Cycle 12

in 1

Chapter 1

Introduction

Consider the system of ordinary differential equations on the plane:

( dx dt=P(x,y) (1.1)

where P, Q are real polynomials in variables (x, y) G M2 without constant terms. Also assume that the system

P(x,y) = 0 (1.2) Q(x,y) = 0 has an isolated solution at (0,0) € K2. We say that (0,0) is a center for system (1.1) if any solution of (1.1) with initial value in a small neighborhood of (0,0) is a closed curve around (0,0). The center problem is to describe all pairs of polynomials P, Q for which (1.1) determines a center. A generalized center problem is similar but with P, Q from some classes of functions (e.g., analytic, piecewise-smooth, etc.). The system (1.1) with P, Q being polynomials is called a planar polynomial vector field. The purpose of this thesis is to give a comprehensive description of some meth ods being used to solve the generalized center problem, and to introduce and classify some new families of polynomial vector fields having centers. The classical center problem was originally posed by Henri Poincare. The original interest of Poincare came from the problem of stability of solar systems in infinite time. Since then, other important applications have been found. At present, some fundamental results in the field assist in many modelling applications in the natural sciences: such dynamical 2 systems are inherent in nature and essential to describing chemical reactions, popula tion dynamics, fluid dynamics and shock waves. Also their behavior allows to predict the stability of systems in a natural environment. It is worth mentioning that David Hilbert in his celebrated 1900 millennium address formulated a problem related to planar polynomial vector fields among 23 of the most important mathematical prob lems for the 20th century (the second part of Hilbert 16th problem). According to Smale, "the second part of Hilbert's 16th problem appears to be one of the most persistent in the famous Hilbert list, second only to the Riemann ^-function conjec ture." Because of the relation between the center problem and David Hilbert's 16th problem, second part, this work will start with a historical introduction to Poincare, Hilbert and others who have been instrumental in the field. Biographical data and mathematical contributions will be mentioned so that the reader can understand the historical progression of the field. Further explanation of how the center problem and the second part of the 16th problem are related, will be explained in Chapter 3. This introduction will also provide background definitions needed to understand the theory, and will conclude with a summary of the items to be discussed in subsequent chapters.

1.1 Henri Poincare

Henri Poincare was born in Nancy, France to a father who was a professor of Medicine on April 29th, 1854. Though born to an affluent family, he had a fragile childhood because of disease. After studying at the Lycee in Nancy, which as now been renamed the Lycee Henri Poincare, he attended the Ecole Polytechnique from 1873 to 1875. He studied further at the Ecole des Mines. While working as a mining engineer, he completed his doctoral work under Charles Hermite; he received his doctorate from the University of Paris in 1879. Poincare's thesis was regarding differential equations, and his examiners were somewhat critical of his work. Upon receiving his doctorate, 3

Figure 1.1: Henri Poincare

Poincare was taught analysis at the University of Caen; he remained here until being appointed to the chair in the Faculty of Science in Paris in 1881. Another appointment occurred in 1886 when he was nominated for the chair of mathematical physics and probability at the Sorbonne. His contributions in mathematics are in many different topics not limited to topol ogy, algebra, geometry, analysis, number theory and differential equations. In physics, he studied optics, electricity, telegraphy, capillarity, elasticity, thermodynamics, po tential theory, quantum theory, theory of relativity and cosmology. He is also ac knowledged along side Albert Einstein and Hendrik Lorentz for the discovery of the special theory of relativity. In [34], Schlomiuk describes Poincare as a co-founder of these systems with publi cations [30] and [31]. Part of his motivation for studying the equations was his work on celestial mechanics. In particular, planar polynomial differential systems are a simplification of the framework of the three-body problem. The other co-founder is Darboux. Poincare admired Darboux's work; however, he laments that Darboux's work did not receive as much attention from geometers as it deserved. In order to correct this, Poincare proposed to the Academie des Sciences it as a subject for the 4

Grand Prix des Sciences Mathematiques. Within the field of planar polynomial vector fields, there are numerous mathematical objects which are named after him:

• Poincare-Bendixon Theorem • Poincare Map • Poincare Transformation • Poincare-Pontryagin Theorem

Poincare considered the problem of giving necessary and sufficient conditions for the existence of a center of a planar polynomial system. The Center-Focus problem was studied by Poincare [32] and further developed by Lyapunov [26], Bendixson [6], and Frommer [19]. He gave an infinite set of necessary and sufficient conditions for one of these systems to have a center at the origin. Poincare also showed that the eigenvalues of the linearization of a system at the origin must be purely imaginary for the system to have a center.

Poincare posed two questions regarding dynamical systems which are of interest to the topic of this thesis. One of them is known as the problem of Poincare; the other is the Center-Focus problem.

It should be noted that Poincare is also the creator of algebraic topology [29]. His main contribution is a collection of six papers published between 1895 and 1904. In these papers, he outlined the foundations of the field, homology, and the fundamental group. This allowed him to develop tools to explore three dimensional manifolds like the 3-sphere. It is this object which made Poincare the most famous as it is the fundamental object of the Poincare Conjecture. He stated it in 1904 as follows: "There remains one question to handle: Is it possible that the fundamental group of a manifold could be the identity, but that the manifold might not be homeomorphic to the three-dimensional sphere?... But this question would carry us too far away". In November 2002, Grigori Perelman proved a more general statement to which the Poincare Conjecture was a consequence. Oddly, Perelman turned down all prizes and 5 awards associated with the answering of the nearly 100 year old question.

1.2 David Hilbert

Figure 1.2: David Hilbert

David Hilbert was born in Koningsberg, Prussia (now Kaliningrad, Russia) on Jan uary 3rd, 1862. Hilbert attended the gymnasium in Koningsberg and after entered the University of Koningsberg. Here he studied under Lindemann for his doctorate which he received in 1885. The thesis was named Uber invariante Eigenshaften specieller bindrer Formen, insbesondere der Kugelfunctionen ( "On the invariant properties of special binary forms, in particular the circular functions" ). At Koningsberg, Hilbert made friends with Minkowski. Hilbert was given the position of Privatdozent from 1886 to 1892, Extraordinary Professor from 1892 - 1893, and full professor in 1893. In 1895, Felix Klein, appointed Hilbert to the chair of mathematics at the University of Gottingen. Hilbert is probably most known for his 1900 millennium address at the International Congress of Mathematicians in which he presented an agenda for the 6 new era of mathematicians. In his speech, he listed ten problems; however, twenty- three were later published. These problems held Hilbert's insight towards the shape of mathematics to come.

Hilbert's 16th problem is of interest to us because of its relation to the Center- Focus problem. Interestingly, in [34], Schlomiuk writes about the differences in ap proaches which have been taken on the 16th problem. In [25], Ilyashenko considers the problem from a perspective of foundation theory in which C°° vector fields with compact parameter spaces are analyzed. Schlomiuk notes that bifurcation has a strong relation to the 16th problem; however, Hilbert's description of the problem is regarding polynomial systems not C°° systems, and though the C°° gives us more freedom to work with, there are insights which Hilbert had which one may miss in generalizing the problem.

Hilbert combined the 15th and 16th problem under one statement of which the preamble was "From the boundary region between algebra and geometry, I will men tion two problems. The one concerns enumerative geometry and the other the topol ogy of algebraic surfaces".

The headings for the two problems are:

• Problem 15: Rigorous foundations of Schubert's enumerative calculus. • Problem 16: Problem of the topology of algebraic curves and surfaces.

The second part is stated as follows:

In connection with this purely algebraic problem, I wish to bring forwards a ques tion which, it seems to me, may be attacked by the same method of continuous vari ation of the coefficients, and whose answer is of corresponding value for the topology of families of curves defined by differential equations. This is the question as to the maximum number and position of Poincare boundary cycles (limit cycles) for a dif ferential equation of the first order and degree of the form ^ = Y/X where X and Y are rational integral functions of the nth degree in x and y. 7

Schlomiuk claims that Hilbert directed this question in a geometric manner - as exemplified by his request to find the position of the limit cycles; this can be further corroborated by the grouping together of the 15th and 16th questions as the 15th question is an enumeration theory of numbers with geometrical meaning [34]. The problem as Hilbert posed it is very difficult in general. Thus, the 16th prob lem is often phrased in terms of just showing that the number of limit cycles in a polynomial system of degree n, H(n) is finite.

1.3 Jean Gaston Darboux

Figure 1.3: Jean Gaston Darboux

Darboux was born in Nimes, France on August 14th, 1842; he attended both the Ecole Polytechnique and the Ecole Normale Superieure. While still completing his Ph.D, Darboux published a paper on orthogonal surfaces. Darboux included this work in his thesis, Sur les surfacs orthogonales in 1866. Darboux taught from 1866 to 1881 at the following French institutions: The College de France, The Lycee Louse le Grand and The Ecole Normale Superieure. From 1873 to 1878, he was suppleant to Liouville 8 in the chair of rational mechanics at the Sorbonne. Later he became suppleant to his Ph.D supervisor, Chasles, in the chair of higher geometry at the Sorbonne. When Chasles died in 1880, Darboux was appointed to the chair of higher geometry - he in turn held this position until he died in 1917. In 1870, he founded the Bulletin des sciences Mathematiques. His first contribution to the field was an article in 1878 [16]. Darboux's most known contribution to the field of planar polynomial vector fields is the Darboux Integral. A function is Darboux-integrable if the difference of the upper Riemann sum and the lower Riemann sum of the function approaches zero as the partition size gets smaller.

1.4 Henri Dulac

Henri Dulac was born in Fayence, France in 1870. He graduated from Ecole Polytech- nique in 1892 with his Doctorate. He then taught in Grenoble, Algiers, and Poitiers. Dulac's teaching duties were suspended during the First World War, and he was man dated to serve as an officer in the French army. Dulac has many accomplishments in the field of planar polynomial vector fields. He is also known for a proof which he provided in 1923 which claimed to show that the upper bound of the number of limit cycles in a planar polynomial vector field exists; this proof was shown by Ilyashenko to have an essential gap. Dulac passed away in Fayence in 1955.

1.5 Yulij S. Ilyashenko

Ilyashenko currently teaches at Cornell University in Ithaca, New York. He received his Ph.D. from Moscow State University in 1969. In 1981, he showed that a previous proof by Dulac which claimed that the upper bound on the number of limit cycles of a planar polynomial vector field existed had a gap. In response to this gap, he fixed the original proof of Dulac, making the fmiteness statement to be true. His proof 9

Figure 1.4: Yulij S. Ilyashenko is partially published in a book Lectures on Analytic Differential Equations with S. Yakovenko.

1.6 Nikolai Nikolaevich Bautin

David Hilbert's 16th problem asks whether an upper bound for the number of limit cycles in a planar polynomial system either exists, and if yes, is it bounded by a function of the degree. So far, much work has been put into this problem; however, only the local upper bound for systems of degree 2 has been found. As stated in [20], in 1939, Bautin announced that he had this result, and in 1952 published a complete proof which stated that the maximum number of limit cycles appearing in the perturbation of a center of a quadratic planar polynomial system was at most 3. He also provided an example of a system with 3 local limit cycles to show that the u A :„ „u„ uuunu io oiidip. Bautin worked under Aleksandr Aleksandrovich Andronov and finished his thesis in 1947 at Nizhni Novgorod Nikolai Ivanovich Lobachevski University. 10

Chapter 2

Classical Results

In this chapter, some classical results on planar polynomial vector fields and their associated differential equations will be discussed. First, the original proof of Bautin's theorem will be given. Second, a description of the most constructive theorem in the field, Kapteyn's result for polynomials systems of degree 2 will be given. Third, the Cherkas Transformation will be introduced, and further, we will show how it is related to Kapteyn's result.

2.1 Bautin's Theorem

Bautin's famous theorem states that the maximum number of limit cycles which appear from a perturbation of an equilibrium position of a polynomial system of degree 2 (so-called, quadratic system) is 3. Before we formally state the theorem and provide the proof, we introduce some definitions. We start by introducing the quadratic system in its most general form and will eventually perform some rotations and simplifications to eliminate extraneous coeffi cients. Let

, i+k=2 , i+k=2 i k i k ~^= Yl aikx y =p(x,y), -| = ^ bikx y = q{x,y) (2.1) i+k=l i+k=l

aio agi be such that ^ 0. In the forthcoming section, it will be explained how bio b0i Bautin proved that the maximum number of limit cycles which appear from an equi librium position for all sufficiently small coefficients a^ and 6^- in (2.1) is 3. We notice that there are ten different coefficients in (2.1); thus, it is intuitive to 11 create a ten dimensional Euclidean space E = M10 to parameterize these coefficients. Bautin's theorem will be formally given after some definitions are introduced. Definition 2.1.1. A singular point of a planar polynomial differential system of type (2.1) is a point (x,y) such that p(x,y) = q(x,y) = 0. Notice that in (2.1), (0,0) is a singular point. Definition 2.1.2. A center is a singular point (XQ, yo) of a planar polynomial differ ential system of type (2.1) such that there exists a neighborhood U of (x0, y0) so that every point in U other than (x0, yo) is nonsingular and the integral curve through the point is closed.

/ ,^^^"->*-*.--4- /

1 1 / V \ ? I I! f f i I ! i .i{.; -F+4- • i » \ \ \ \ \- '/ / / / "I I i ~^-s/ / / / W'v^---—- -fSS / / / -^•S'J' / / / -~-^s's'S / \\\v.vv

Figure 2.1: A Vector Field with a Center

Definition 2.1.3. A focus is a singular point (xo,yo) such that Ve > 0, 3d > 0 such that the trajectory (x(t),y(t)) of a vector field passing through a point of Bs(xo,yo) satisfies

lim (x{t),y(t)) = (x0)yo) I—• — OO and 3T0 (depending on this trajectory) such that Vi > T0, (x(t),y(t)) £ Bt(xo,yo) where Br(x0, y0) is the closed disk of radius r centered at (x0, y0). Definition 2.1.4. A local limit cycle is an isolated closed trajectory around a singular point. 12

Figure 2.2: A Limit Cycle

We will proceed with the definition of cyclicity, see also [5] for its definition. Definition 2.1.5. Let a = (aio, • • • ,^02) G R10. The equilibrium position (x,y) = (0,0) of a system (2.1) with the coefficients above satisfying

aio&oi - aoifrio > 0 and aw + b01 = 0 has cyclicity of order k > 0 with respect to E if

(a) 3eo,5o > 0 such that for a G E,$b S Beo(a) such that the system corresponding 2 to b has more than k limit cycles in B§o(0) C M .

(b) Ve < eo,S < 60, 3b £ Be(a) such that the system corresponding to b has k limit 2 cycles in Bs(0) C M .

Here Bs(a) is the open disk in M10 of radius r and center a.

Bautin's theorem states that

A singular point of center or focus type of a quadratic system has cyclicity less than or equal to 3 with respect to all quadratic systems.

The sharp form of Bautin's theorem is the subject of Theorem 2.1.8 below. We now simplify the parameter space of the differential system (2.1). In fact, it is possible to transfer our system (2.1) into a system with less parameters. Specifically, the system 13

(2.1), subject to the conditions

2 a-wboi - aoibw > 0 and (aw - b01) + 4a0i&io < 0 can be transformed by means of the substitution

x = — [(al0 - 01)77 + &i£] and y = r7? hb 10 Ol to the canonical form

drj 2 2 = ht + axr) + B20£ + £n£r/ + B02T]

-£- = cut - hr) + A (2 + Auto + A rj2 (2.2) at i 20 02

where ax and b\ are the real and imaginary parts of the roots

A*i = ai + b\i and A*2 = «i — M of the characteristic equation

aio — A* °oi = 0. &io &oi - fJ-

We can rotate the coordinate system so that B2o + B02 = 0. Since the discriminant of the characteristic equation is negative by assumption, Im(/Xj) 7^ 0, thus bi ^ 0. So we can divide (2.2) by 61 to get the following equations

(IT (2.3) 2 2 — = Aix - y - X3x + (2A2 + A5)xy + A6y = P(x, y)

2 2 •J = x + Aiy + A2x + (2A3 + A4)xy - \2y = Q{x, y). 14

Note that we have set Xi — f- and t = b\ti. Also, observe that when Ai = A4 = A5 = 0, we get the following

P!c(x,y) = -2X3x + 2X2y

Q'y(x,y) = 2X3x-2X2y.

Thus P'x + Q'y = 0. Thus, under this restriction, we have a center type singularity at the origin.

We also note here that if (2.1) has cyclicity of order k, then (2.2) also has cyclicity of order k as the only difference between them is a reparameterization.

We now transfer our system using polar coordinates (p,

2 2 3 x = —p + p cos 4> = \\p cos 4> — psincj) cos (f> — X3p cos (2.4)

2 2 2 2 + (2A2 + A5)p cos 0 sin 0 + Xep sin cj> cos (f>

2 2 y = p cos 4> + p sin 4> — P cos 0 + Aip sin (f> + X2p cos

2 2 2 + (2A3 + A4)p cos (j) sin — A2p sin cf>.

Now, multiplying (2.4) by cos, and (2.5) by sin0, then adding these two gives us p, and multiplying (2.4) by sin and (2.5) by — cos and adding gives us 0 as written below:

3 2 p =X\p-—- A$p cos- - + (3A2- + As)p cos—0sin <£

2 2 2 3 + (2A3 + A4 + A6)p cos 0 sin - A2p sin 0

3 2 0 = 1 + A2 cos 4> + (3A3 + Xi)p cos 0 sin (/»

2 3 — (3A2 + X5)p cos 4> sin 0 — A6p sin

Since both equations are written in the same independent variable, say t, we can write

dp_=P_ dcj) $

Further, -& can be expanded into a series according to powers of p and some polyno mials Rk where the Rk are polynomials of sin 0, cos 0, and Aj. This series

2 3 ^=pR1 + p R2 + p R3 + ... (2.6) converges for all 0 and for all Aj such that

\Xi - A*| < e in the neighborhood of an arbitrary point A* of E for all sufficiently small values of p, such that \p\ < r(e, A*). The solution p = p(

u p = pov^cf), Xt) + plv2((j>, A*) + po s(0, Aj) + ... (2.7) converging for all 4> such that 0 < < 2TT, for all values of Aj such that |Aj — A*| < e and for p0 such that \p\ < r(e, A*). Since p(0) = p0, we have

ui(0,Ai) = l, vk{0,Xi) = 0, fc = 2,3,....

Substituting the series (2.7) in the equation (2.6) and comparing coefficients for in dividual .powers of powe obtain, for the determinationof thecoefficientsof -£>/-, -the- 16 recursive differential equations

dvi le V1R1, dv 2 v Ri + v\R , ~dl 2 2 dvz v Rx + 2v v R + vfR , de 3 1 2 2 3

from which we determine the Vk as integral functions of

2 p = POVI(2TT, Xi) + p 0v2(2ir, A*) + p^3(27r, A*) + .... (2.8)

Now we state two technical lemmas which are proven in Bautin's original paper. These lemmas will only be stated here, see [5] for their proofs. Lemma 2.1.6. The coefficients Vk(2ir, Xi) for k = 1,2,... and i = 1,2,... ,6 of the above sequential function are integral functions of the parameters Aj, which for Ai = 0 become homogeneous polynomials of degree k — 1 in the parameters A2, A3, A4, A5, A6- 17

Lemma 2.1.7. The coefficients Vk(2n, A,) have the form

2vX v1(2ir,Xi) = e \

1 v2(2ir,\i) = \4 \

{ ) v3(27r,Xi) = v^ + X1e 3 ,

) v4(2TT,Xi) = nef + X19^\

3) 1) W5(27r,Ai)=^ + ^ + A1^ ,

3) V6(2TT, Xt) = U# + ^ + Ai^,

3) v7(27r,Xi) = v^ + vE6<7 + X18?\

(7) ) , ^-a(5) , 51-/1(3) ) , , ,(i) vk(2ir,xi) = v^e]; + vEe^+we^ + x1e];' for k>7,

where

7T w3 = --rA5(A3 - A6),

w5 = ^TA2A4(A3 - A6)(A4 + 5A3 - 5A6),

2 2 W = ^-WA - 3 - A6) (A3A6 - 2X 6 ~ %)

and the 9^iCi ) are integral functions of the Aj. Now, we formulate the complete version of the Bautin theorem. Theorem 2.1.8. A system (2.1) can only have equilibrium positions of focus or center type whose order of cyclicity with respect to the space E = M.10 is 0,1,2, or 3, and does not have equilibrium positions with an order of cyclicity greater than 3.

Proof. We consider the function

P ~ Po = [vi{2ir, Xt) - l]p0 + U2(2TT, A*)^ + U3(2TT, A^PO + ..., (2.9) 18 whose positive zeros correspond to limit cycles. Using Lemma 2.1.7, we represent (2.9) in the form

p- Po = A){2TTAI[1 + Ai^i(Ax) + poii>i(po, \)] + v3[l + Pofaipo, \)]pl (2.10)

+ v5[l + p0ip(po, Xi)]pQ + v7[l + po^r(po, K)}po} or alternatively in the form

P ~ Po = po[2irXiipl + v3ip*3pl + v^lpl + v7^7pl], (2.11) where ip*, defined for j = 1,3,5,7, are series of powers of po> whose coefficients are integral functions in Aj. These series converge inside a neighborhood \Xt — A*| < e of an arbitrary point Aj G E and for all p < r. We show that condition (a) from Definition 2.1.5 can be satisfied by the system (2.1). For this we select an e-neighborhood of the point A, in the coefficient space of the system (2.3) and consider two cases. Case 1. Let all the ffc(27r, A^) = 0 at the point A, in question (this is equivalent to the case where (2.3) has a center at the origin). It is then possible to find numbers ei and <5i(ei < e) such that |Aj — A*| < t\ and 0 < p < Si hold the conditions

^*>\ for j = 1,3,5,7.

We represent (2.10) as follows

P ~ Po = Pofl ^A1+%|^+%Jk+»7|k for 0 < p < Si. For sufficientlsufficiei y small p we can again represent each of the ratios -^ in the form of an integral series of powers of p0 and A,, with coefficients which are integral series in 19 the remaining parameters as follows

) ) ^ = l + X1^ +p0^ =rj* for j = 3,5,7.

We can now find e2 and 62 such that for all Aj in an e2-neighborhood of the point

A* and for 0 < p < S2 there hold the conditions

1. ^T > 2*"•

The positive zeros of the function p — p0 for 0 < p < 52 will coincide with the zeros of the function

2 ^0 = 2nX1 + v3r3*p 0 + v^Po + WPo- (2-12)

We have

5 5 3 Vop = 2u3(l + Ai^ + Po^iVo + u5(l + A^g + Po4 i)p o ) ) 5 + 6v7(l + Xlg ) + p04l )p 0.

We observe that for 0 < p < 52 the number of the positive zeros of the function (2.9) or, what is the same thing of the function (2.12), cannot exceed the number of positive zeros of the function

5 i> = 2tJ3(l + \i$ + Po^i) + 4U5(1 + Ax^ + po4 i)pl (2-13)

7 ) + 6^7(l + A1(/>2 1 + w/'2?)Po 20 by more than unity. We choose now £3 and €3 such that each side of the ratios

1 + Ai^/ + p0^21 for j = 5,7, for all Aj in an e3-neighborhood of the point A* and for 0 < p < 5s satisfies the condition

r;*>\ for j = 5,7.

Then for 0 < p < 53, the positive zeros of the function (2.13) will coincide with the zeros of the function

fo = 2v3 + 4y5^rVo + 6u7^*Vo-

We have

5 WoP = 8^(1 + M4 I + Po4f)Po + 24^J7(l + \l(j><£ + po4^)Po-

The number of positive zeros of the function (2.13) for 0 < p < 53 cannot exceed the number of positive zeros of the function

7 ^ = 8t75(l + Ai^? + Po^?) + 24U7(1 + Xtfg + po4 iVo (2-14) by more than unity. Continuing this process a further step, we obtain:

^ = 48tJ7(l + Axri? + po^i?), (2.15) where the number of positive zeros of the function (2.14) for 0 < p < 5^ and all 5i in an e4-neighborhood of the point A* cannot exceed the number of positive zeros of 21 the function (2.15) by more than unity, so that the number of positive zeros of the (2.9) cannot exceed the number of positive zeros of the function (2.15) by more than 3. However there obviously exists for e$ and 65 such that 0 < p < 64 and all Aj in an e5-neighborhood of the point A* the function (2.15) which has no zeros. From this it follows that for all 5t in an e0-neighborhood of the point A* and for 0 < p < 80, where eo = minjej : i = 1, ...,5} and 5o = min{5j : i = 1, ...,5}, the function (2.9) has not more than three positive zeros.

Case 2. At the point A, in question of the coefficient space let not all the Vfc(27r, A,) be zero. The system (2.2) has a focus equilibrium position at the origin. Let for example, t>3(27r, Aj) be the first non-zero coefficient of the function (2.9). Then, it is possible to represent the function given in (2.9) in the form below

p - p0 = PO{2TT\1 [1 + Ax0i + po^i] + [V3 + Po^z] pl}- (2.16)

It is also possible to find an ei-neighborhood of the point A* such that within this neighborhood Vz 7^ 0. We can also choose ti and 5? such that for 0 < p < 82 and all A, in an e2-neighborhood of the point A*, the following conditions hold

1 _ 1 1 + Ai^i + pV>i > r and v3 + Poi>z > 2> and represent (2.16) as

p - p0 = p0 [1 + Ai^g + p0ipi] {2?rAi + [v3 + Ai0* + p0^*] pl}- (2.17)

For 0 < p < 52, the positive zeros of the function (2.17) coincide with the zeros of the function

•0* = 27rAi + [v3 + Ai^* + pip*} p\. 22

We also have

tf = 2p0 [v3 + Ai0„ + poA*}, (2.18) where the number of positive zeros of the function in (2.18) for 0 < p < 82 cannot exceed the number of positive zeros of the function in (2.16) by more than unity.

Hence, just as before, it follows that we may find eo and 5Q such that for 0 < p < 5Q and all Aj in an e0-neighborhood there is not more than one zero of the function (2.9). It is proved in a similar way that, in the case in which the first non-zero coefficient of the function (2.9) is V5(2n, Aj), it is possible to find do and eo such that for 0 7(27r, Aj), for 0 < p < S0 there are not more than three zeros of the function (2.9).

We now consider part b) of Definition (2.1.5) can be satisfied for the system (2.1). We consider once more the function (2.9), and suppose the point in question A, of the coefficient space to be such that v(2n, Aj) is the first non-zero coefficient of the function given in (2.9). For definiteness, let u7(27r,Aj) > 0. Whatever e < e0 and 5 < 50 are, the conditions ipj > | (for j = 1,3,5, 7) will be satisfied for all Aj in an e-neighborhood of the point A* for 0 0 is obviously satisfied for the point A* in question (with Ai = A5 = A4 + 5A3 — 5A6 = 0) and for 0 < p < 5. We will now vary A4 within the e-neighborhood so that v$ becomes negative, and at the same time so small that for all p, satisfying the condition 5$ 0.

Since for sufficiently small p, the sign of p — p0 will coincide with the sign of v5 (v3 does not depend on A4), there follows from this the existence of at least one zero of the function p — po in the interval (0,(5). Let p = p' be the least of the zeros. We shall now vary A5, leaving the remaining coefficients fixed, and remaining as before inside the e-neighborhood of the point A*, in such a way that v3 becomes positive, 23 and so small that for all p satisfying the condition 5\ ^2 ) *s some interval contained in (0,p'), there holds the inequality p — p0 < 0 (v5 and Vj do not depend on

A5). For sufficiently small p the sign of p — p0 will coincide with the sign of A4; from this there follows the existence of at least two zeros of p — p0 in the interval (0,5). Let p = p" be the least of these zeros. We shall now vary Ai (making it negative) in such a way that:

1) for all p which satisfy the condition <5g < p5% , where (£3 ,<53 ) is some interval

contained in (SQ ',5Q ), there holds the inequality p — p0 > 0.

2) for p satisfying the condition 5\'

contained in (S[ ,5{ ), there holds the inequality p — p0 < 0.

3) for p satisfying the condition 55' 0.

For sufficiently small p, the sign of p — po will coincide with the sign of Ai. From this there follows the existence of at least three zeros of p — p0 in the interval (0,5). For points A* of the coefficient space for which the first non-zero coefficient of the function (2.9) is v$ or VT, it is possible to prove in a similar way that there exists in an e-neighborhood of such a point, a point such that the corresponding to it system (2.2) has two or one limit cycles in a ^-neighborhood of the origin. It is obvious that for points of the coefficient space in which Ai 7^ 0, the equilibrium position (x, y) = (0,0) has, with respect to the space of the Aj, cyclicity of order 0. The proof of Bautin's theorem is complete. • 24

2.2 The Center Conditions for Degree 2 Planar Polynomial Systems

The history related to the study of quadratic vector fields is described in [34],[15]; a summary is given here. The theory of planar polynomial vector fields was initiated by Poincare and Darboux at the end of the 19th century. Notably, one establishing article was written by Darboux in 1878, and two were written by Poincare in 1881 and 1885. The concept of the center was defined by Poincare in his first article. He was also the first to consider necessary and sufficient conditions for the existence of a center given by a system

x = P(x,y)

y = Q(x,y) where P and Q are polynomials. Poincare gave an infinite set of necessary and sufficient conditions for these systems to have centers at the origin. In 1947, Lya- punov published a paper on systems of differential equations in n variables. When Lyapunov's results were applied to polynomials of degree 2, another infinite set of necessary and sufficient conditions for the existence of a center at the origin was created.

One necessary and sufficient condition which was published by Poincare in 1881 is contained in the following theorem Theorem 2.2.1. A necessary and sufficient condition for the analytic system

x = ax + by + ^ Xk(x, y) fc=2 oo

y = ex + dy + ^Yk(x,y) k=2 25

where the coefficients a, b, c, d are such that the matrix has purely imaginary c d eigenvalues, to have a center at the origin, is that the system has a constant of the motion F(x, y) in a neighborhood of the origin, where F is real analytic and not constant. The interesting part, demonstrated by Poincare in 1881, is that the condition - that the eigenvalues of the above matrix be purely imaginary - is necessary for the system above to have a center. Dulac was the first to study the center problem for quadratic polynomial vector fields in 1908. In his publication, Dulac lists all possible cases of quadratic polynomial vector fields having a center at the origin with complex coefficients; Dulac does not consider the case of the center of quadratic polynomial vector fields over the reals. In 1911, Kapteyn gave a set of algebraic conditions on the coefficients of a real quadratic system of differential equations for the system to have a center at the origin. These conditions were only sufficient since not all quadratic systems were considered. After studying Dulac's paper published 3 years earlier, Kapteyn extended his method to the complex quadratic differential systems and obtained necessary and sufficient algebraic conditions, in terms of the coefficients of the system, to have a center. If these results are restricted to the real vector fields, the following theorem is obtained. Theorem 2.2.2. Every quadratic system,

x = —y — bx2 — (2c + /3)xy — dy2,

y = x + ax2 + (26 + a)xy + cy2

can be brought by a rotation to a system of the form with a + c = 0. Moreover, assuming that a + c = 0, a necessary and sufficient condition for such a system to have a center at the origin is to satisfy one of the following conditions: . I) b + d = 0 26

II) P = 0 = aa

III) p = 0 = a + 5(6 + d) = a2 + 2d2 + 6d

In [15], an outline of Kapteyn's proof stemming from Bautin's work is given. The proof that every such system can be brought by a rotation so that a + c = 0 is immediate. The proof of the biconditionals is proved by first showing sufficiency of I,II or III and then using the sufficiency and a theorem by Bautin in [5] to show necessity.

2.3 Cherkas Transform

As shown in [7], in the specific case of vector fields with homogeneous polynomials F and G as below, one can reduce the Center problem for the system

- = -y + F(x,y)

ft=x + G(x,y). (2.19) to a similar problem for the Abel differential equation by means of the so-called Cherkas transform. The latter equation seems to be much simpler than the original system, giving a hope of a complete solution of the center problem in this case. For the general case, we will consider equations in (2.19) such that F(x,y) and G(x, y) are homogeneous polynomials of degree d. If we write (2.19) in polar coordinates with x = rcos9,y = rsin#, then we get

x = r cos 9 — r sin 9 • 9 = — r sin 9 + F(r cos 9, r sin 9) (2.20)

y = rsind + rcos9 • 9 = rcos9 + G(rcos9,rsin9). (2.21) 27

Multiplying the first equation by cos#, and the second by sin# and adding them gives

r = F{r cos 9, r sin 9) cos 9 + G(r cos 9, r sin 9) sin 6. (2.22)

Since F and G are homogeneous of degree d, we can write

d r = r (F(cos 9, sin 6>) cos 9 + G(cos 9, sin 9) sin 0)

= r"/(0).

Similarly, by multiplying (2.20) by — sin# and (2.21) by cos# and adding them, we get

r9 = r- F(r cos 9, r sin 9) sin 9 + G(r cos9,r sin 9) cos 9 (2.23) and simplifying, we get

d 1 0 = l + r - (-F(cos0,sin0)sin0 + G(cos0,sin0)cos0)

= 1 + rd-1g{9).

If we combine these equations we get

dr _r _ rdf(9) d~9~ 9 ~ l + rd-lg{9)'

Then, we apply the transformation suggested by L. Cherkas

rd-l

P= l + rd-lg{9Y

To simplify the algebra, let A = 1 + rd~1g(9). Then, taking the derivative of p with 28 respect to 9 gives

dp 1 A(d - l)/"2^ - r^ (r^g\9) + (d - V"2^^) d9

Now substituting 45 = r -.' ' gives

dp _ 1 A(d - IY -T - rr d-i [ r° lrff(9)d-ui + (d- l)rd-2-jg{9) d9~A^ „d-l\2 ,d-l\ 3 ((d-l)f(9)-g'(9)) -f(9)g(9)(d-l) A ((d-l)f(9)-g'(9))p-f(9)g(9)(d-l)p.

Thus, if we leftp(0) = (d-l)f(9)-g'(9), and q{9) = -(d-l)f{9)g{9) by polynomials in sin#, and cos 9, of degree d + 1 and 2d + 2 respectively, we get

3 ^ = P(

Thus, it is shown that a differential system of the type (2.19), under the condition that F(x, y) and G(x, y) can be reduced to the Center problem for the Abel equation (2.24). Now, the trajectory of (2.19) near the origin is closed if and only if the corresponding solution of (2.24) satisfies p(0) = p(2ir). Thus, the solution is periodic. Therefore the center problem for (2.19) is translated into the problem of finding conditions on p and q for which all solutions of (2.24) are periodic. We will consider a model for the center problem for Abel differential equations. 29

2.4 Abel Equations Obtained from Vector Fields of Degree 2

In this section, we are interested in applying Kapteyn's result Theorem 2.2.2 to Abel differential equations. The method will be an elementary change of variables through out the translation in order to find the characteristics of p(8) and q(9) which create a center. First, recall that

p(9) = (d-l)f(9)-g'(9) and q(e) = -(d-l)f(9)g(9).

Thus, working backwards me must obtain / and g; since we are only considering degree 2 (d = 2) systems here, instead of using the notation F(x, y) and G(x, y) stated previously, we write:

F{x, y) = -bx2 - (2c + /3)xy - dy2

G(x, y) = ax2 + (26 + a)xy + cy2.

But by Theorem 2.2.2, this system may be rotated such that a + c = 0. Thus we can eliminate a. From the previous section, we have

f(6) = F(cos 9, sin 9) cos 9 + G(cos 9, sin 9) sin 9

g{0) = -F(cos 9, sin 9) sin 8 + G(cos 9, sin 9) cos 9. 30

These can be expanded as we must also know g'(0) and f(9)g(9) in order to calculate p(9) and q(9).

f(0) = -bcos36- (3c + p) cos2 0sin0 + (26 - d + a) cos 9 sin2 0 + csin30

3 2 2 3 0(0) = -c cos 6> + (36 + a) cos (9 sin 0 + (3c 4- /3) cos 9 sin 0 + d sin 0.

The following are also given here

g'(d) = (36 + a) cos3 0 + (96 + 2/3) sin 0 cos2 0

+ (-66 - 2a + 3d) sin2 0 cos 9 + (-3c - (3) sin3 0.

f(9)g(9) = 6c cos6 0 + (uc - i>6) sin 9 cos5 0 + (-wc - uv - bu) sin2 0 cos4 0

= +(—c2 + iwu — u2 — bd) sin3 0 cos3 6 + (cv + wu — ud) sin4 0 cos2 0

= +(cu + u;d) sin5 0 cos 0 + cd sin6 0,

where

« = 2c + f3

v = 36 + a

w = —d + 26 + a.

From this, we can deduce p(9) and q(9) for systems of degree 2. We will write them down explicitly for each case of Kapteyn's result in Theorem 2.2.2. 31

Case 1, b + d = 0.

p(9) = -{4b + a) cos3 0-(12c+ 3(3) cos2 6sin6

+ (126 + 3a) cos 0 sin2 0 + (4c + (3) sin3 0,

q(9) = —6c cos6 0 — (ttc — f 6) cos5 9 sin 9 + (vc + uv + ub) cos4 9 sin2 0

+ (c2 - u2 + it2 - 62) cos3 0 sin3 9 - (vc + vu + u6) cos2 0 sin4 9

— (uc — w6) sin5 9 cos 0 + 6c sin6 9.

Making appropriate simplifications, these become

p(9) = -A cos3 9-W cos2 9 sin 9 + 3A cos 0 sin2 0 + 5 sin3 0,

6 5 4 2 g(0) = -C cos 0 - D cos 0 sin 0 + E cos 0 sin 0

- (-c2 + -u2 - u2 + b2) cos3 0 sin3 0 - £ cos2 0 sin4 0 - D cos 0 sin5 0 + C sin6 0.

Though the above result exhibits the symmetry of the system, it may be misleading, as it suggests that the new variables A, B,... etc. are not related to a, (3,... etc. This is however not the case, and the following representations must be taken into account.

A = 46 + a

B = 4c + P

C = bc

D = 3c2 + (3c - 362 - ah

E = 4(3b + 4ac - 9c6 + /3a. 32

We can actually further eliminate as E = AB — 25C; thus,

p{$) = -A cos3 9 - 3B cos2 9 sin 9 + 3A cos 9 sin2 9 + B sin3 9,

q{B) = -C cos6 0 - D cos5 9 sin 0 + (.45 - 25C) cos4 0 sin2 9

- (-c2 + u2 - u2 + 62) cos3 9 sin3 0

- (AB - 25C) cos2 0sin4 9-Dcos9sin5 0 + Csin6 0.

Case 2, /3 = 0 = act.

We break this case into two subcases as the condition that aa = 0 could imply that either a = 0 or a = 0.

Case 2.1 /3 = 0 = a = -c.

p{9) = (-46 - a) cos3 9 + {-4d + 8b + 3a) sin2 0 cos 9,

q(9) = 6(36 + a) sin 0 cos5 0 + ((d - 26 - a)(3b + a) + bd) sin3 0 cos3 9,

- (-d + 2b + a)dsin59cos9.

Case 2.2 /? = 0 = a

p(6) = -46 cos3 9 - 12c sin 0 cos2 9 + (86 - 4d) sin2 0 cos 0 + 4c sin3 9,

q(9) = -6ccos6 0 + (-3c2 + 362) sintfcos5 9 + (146 - d)csin2 0cos4 9

+ (4c2 + Abd - 662) sin3 9 cos3 9 + 3c(2d - 36) sin4 9 cos2 0

+ (-3c2 - {-d + 2b)d) sin5 0cos9 - cdsin6 9.

Case 3 (3 = 0 = a + 5(6 + d) = a2 + 2d2 + bd.

This case is more complicated as there are squares in the restriction given by The orem 2.2.2. Thus, we will create a new variable in order to simplify our expressions. 33

We make the following substitutions considering that a + c = 0.

0 = 0

a = —5b — 5d

c2 = -d(2d + b)

G = c to get the following conditions on p(9) and q(9):

p(0) = (b- 5d) cos3 9 - Y1G sin 9 cos2 9 - (7b + 19d) sin2 9 cos 9 + 4G sin3 9,

q(9) = -bG cos6 9 + {-2b2 - 2bd + 6d2) sin 9 cos5 0 - G(6b + 21rf) sin2 0 cos4 9

+ (-6b2 + 64bd - 50d2) sin3 9 cos3 9 + G(Ub + 26d) sin4 9 cos2 0

+ 6(bd + 2d2) sin5 9cos9- Gdsin69. 34

Chapter 3

Algebraic Methods

3.1 Introduction

In this chapter, we describe some modern algebraic and topological methods devel oped recently in connection with the generalized Center problem. We start with an explicit expression for the first return map for ordinary differential equations obtained in connection with the Center problem and then describe a natural class of centers satisfying some topological conditions.

3.2 An Explicit Expression for the First Return Map

Poincare found necessary and sufficient conditions for the origin to be a center of a planar polynomial vector field. Poincare's conditions are recursive, and thus, com putationally obtuse. In [8], with respect to the generalized Center problem, explicit formulation for these conditions are given. We investigate the proof here. As seen in the Cherkas Transform section, the polynomial differential system

x = -y + F(x,y)

y = x + G(x,y)

with F, G without constant and free terms can be transformed into

dr ^ f(r,0) d9 1 + g(r, 9) 35 where f(r, 9) and g(r, 9) are polynomials of degree d in r whose coefficients are trigonometric polynomials in 9, d is the degree of the planar polynomial vector field, x = rcos9,y = rsin9. Moreover / does not have free and linear terms in r, and g does not have free terms in r. So we can express the right-hand side of the above equation as a series in r:

^ = 5>'+1 (3,) where aj are trigonometric polynomials in 9. We will consider a more general case of equation (3.1)

, OO l+1 fx=Y,^)P (3-2) i=l with a,i bounded measurable complex-valued L°°(S1) 27r-periodic functions and S1 := M/27rZ as the unit circle. If the coefficients of (3.2) do not grow fast for a sufficiently small initial value, one can solve the equation by Picard iteration to obtain a locally Lipschitz solution of (3.2) on [0,27r]. An extended definition of a center of equation (3.2) for sufficiently small initial values p(0) is that p(0) = p(2n). That is to say all solutions of (3.2) with sufficiently small values are 27r-periodic. Our goal will be to find conditions on the coefficients of (3.2) under which this equation determines a center. For important results related to these see [2], [4], [1], [3], [12], [14], [17], [23], [24], [35], [33], and [37].

1 We make the following assumptions and conditions: Let Xt = L°°(S ) be the space of all coefficients a,i and X be the complex Frechet space of sequences a =

(ai, a2,...) € U^Xi such that

sup \cii(x)\ < l{a)\ i = 1,2,..., xeS1 36 for some 1(a) G M.+ . Definition 3.2.1. Let C C X denote the center set of (3.2); in other words, C is the set of a G X for which the corresponding equations (3.2) determine centers. Definition 3.2.2. Consider the iterated integrals

a s hu...,ik{a) = •• ik( k) •' • Oii(si) dsk • • • dsi J Jolinear holomorphic functions on X. A linear combination of iterated integrals of order < k is called an iterated polynomial of degree k. Definition 3.2.3. For a sufficiently small r, let v(x; r;a),x G [0, 2TT] be the Lipschitz solution of equation (3.2) corresponding to a G X with initial value v(0; r; a) = r. Then, P(a)(r) = v(2ir; r; a) is the first return map. Theorem 3.2.4. For sufficiently small initial values r, the first return map P(a) is an absolutely convergent power series

oo n+l P{a)(r) = r + ^ cn(a)r where

71=1

Cn{d) = ^ Ch ' • ' ' ' CiJiu-Jk (a) and H+.-.ik—n

Ciu...,ik = {n - h + 1) • (n - ii - i2 + 1) • (n - ix - i2 - h + 1) • • • 1.

The center set C C X of equation (3.1) is determined by the system of polynomial equations Ci(a) = 0, i = 1, 2,....

Proof Into (3.2) substitute tp(x) instead of p(x) then we have

i i+l V'

Multiplying this by pk 1 and using the chain rule, we get

oo (vhY = $>Oi(z)*V+*. (3.4) t=i

Let V be the linear space spanned by the vectors e$ = (81^,82,1, • • •) where 8 is the Kronecker symbol. 8ij = 0 if i 7^ j and 5^ = 1. We set y$ = v.'ej and K = (yi, y-z,...). Combining equations (3.4) for all k we obtain the following system

Y'=(JTAMX)AY- (3-5)

Here Aj : V —> V are linear operators with V identified as the algebra C[z] of

n l formal power series with complex coefficients so that en coincides with z ~ . Let D,L : C[z] —> C[z] be the differentiaion and the left translation operators defined on /W = E^ocfc2;fc by

00 00 k k (Df)(z) = J2(k + l)ck+1z and (Lf)(z) = J^ck+1z . (3.6) fe=0 fe=0

Lemma 3.2.5. It is true that

Ai = DV'1.

Thus (3.5) acquires the form

Y' = ( ]C a^fDU-1 J y. (3.7)

The above lemma is given without proof as it can be shown explicitly by computing DU~l. Let u;(x) = ^^ a^x)?DLl~l. The fundamental solution Y of (3.7) can be 38 obtained by Picard iteration to get

Y(x) = I + ^2 / '"' / w(sn) • • • w(si) dsn-- • dsi n=l J J0

Iiu...,ik(a;x)= ••• aik(sk) • • • ah(si) dsk • • • dsi, iGi J Jo

These equations can be rewritten as

Y(x) = I+ ^2pi(x;D,L)ti where (3.8) i=l I i 1 h 1 Pi(x;D,L)= J2 iu-,ik(a;x){DL "- )---(DL - ), i > 1. (3.9)

By definition, the solution Y(x;z,t,r) of (3.7) with initial value Y(0) = X^So r%+1 z% is Yl'iLoy(.x'>t,T)1+1Z1 where y(x;t,r) is the solution of (3.2) which has initial value y(0;t,r) = r. Thus, P(a)(tr) := ty(2ir,t,r). Thus, in order to compute the co

efficients of the first return map P(a)(r), one should apply every pn(2ir, D, L) to l+1 % X^o r z and then substitute z = 0. We get the same if we apply every pn(2-rr, D, L) to zn. From here and (3.9), we get i\ + • • • + ik = n,

1 l n du...tik := {DL^- ) • • • {DL^- )(z ) = (n - ix + 1) • (n - h - i2 + 1) • • • 1.

Therefore

oo n+1 P(a)(r) = r + ^2cnr where cn(a) = ^ ciu„.tikIiu..Jk(a). n=l iiH hifc=n

D 39

Corollary 3.2.6. (a) cn(a) = In(a) + fn(a) where In(a) := f * an(s) ds and fn is an iterated polynomial of degree n in a\,..., an_i; (b) The set

Cn = {a € X : ci(a) = c2{a) = ••• = cn(a) = 0} is a closed complex submanifold of X of codimension n containing 0 e X;

/1(a) = --- = /„(a) = 0.

Proof (a) follows directly from the definition for cn(a). That is,

c a c a n\ ) — / J ii,...,ik^h,--,ikK ) il+...+ik=n c l = / _, h,...,ikh\,...,ik\( ) + cnln\a). ii+...+ik=n-l

But cn = 1, thus we have

cn{a) = fn(a) + In(a)

(b) for this proof, we will define a map -K^ : X —> HJ=1XJ; this is denoted as the natural projection to the first k coordinates. According to part (a), the set Cn is described by the equations Ik{o) = —fk{a) for 1 < A; < n, where fk depends on a^ and Ik depends on ai,... a^-i- Now let

Cn = {ae irn(X) : 4(a) = -/fc(a), 1 < k < n}. 40

1 We have that Cn = 7rn (Cn). Thus, for our purposes, it suffices to show that

Cn C Hj=1Xj is a closed complex submanifold of codimension n containing 0. We will prove this by

W induction on n. If n = 1, thenCi is determined by the equation A (a) := J0 a\(s) ds = 0. Since I\ is a continuous linear functional, C\ = ker i^. Thus the required statement holds. Suppose that the statement holds for k — 1, we will prove the statement for k. Easily, we see that

Ck C Cfc_i x Xk

Since Ik\xk is a continuous linear functional, we can decompose Xk = Ek © /&, where

Ek = ker(JfcUfc)

lk = {vek : v e Cand/fc(efc) = 1}

Let us consider the set Rk := Ck_\ x Ek. It is clear that Rk C ker(/fc) and C^_i x Xk =

Rk © 4- So, for any a € Ck-\ x Xk, we have a = w + vek for some w G Rk and u € C.

Thus in the definition of Ck we have

v = Ik(a) = -fk(a) = -fk(w).

This shows that Ck is the graph of the function — fk : Rk —> C But according to the induction hypothesis Rk C (U^lXj) x Ek is a closed complex submanifold of codimension (fc — 1) containing 0. From here it follows that Ck C Ck-\ x Xk is a closed complex submanifold of codimension 1 containing 0. This proves (b).

To prove (c), note the linear term of the McLaurin expansion of cn{a) at 0 6 I equals /„. But the tangent space to Cn at 0 € X is the set of common zeros of 41

the linear terms of c\,... ,cn. In other words, the tangent space is determined by

71(a) = --- = /n(a)=0. •

Theorem 3.2.7. An element a = (ai,a2,...) G X belongs to the center set C if and only if there is a sequence ui, U2, • • • of 2ir-periodic Lipschitz functions such that Ui(0) = 0 for any i and

as a formal power series in t.

Proof. Let v(x;r;a)x G [0,2TT], be the Lipschitz solution of equation (3.1) corre sponding to a = (ai,a2,...) G X with initial value v(0;r;a) = r. Then, from Picard iteration, it follows that

oo v(x;r;a) = r + ^2vi(x;a)ri+1, (3.10) i=l where each Vi(x; a) is a Lipschitz function on [0, 2TT] and the series converges uniformly in the domain 0 < x < 27r, \r\ < r for sufficiently small positive f. Assuming that a G C, we get t>j(0;a) = Vi(2ir;a) = 0 for any i. Now the inverse function theorem implies that there is a function u(x; p; a) := P + Y^HLI ui{x'i a)pl+1i where each Ui(x; a) is a 27r-periodic Lipschitz function Uj(0, a) = 0, and the series converges uniformly in the domain —oo < x < oo, \p\ < p for sufficiently small positive p, such that

u(x;v(x;r;a); a) = r for all sufficiently small r. Differentiating the function in x we get

< u x a v x r a k+1 dv(x;r;a) = Tl k=i 'k( ^ ) ( '^ i ) dx l + Y^°=i(k+l)uk(x',a)v(xir',a)k' 42

Using the equality v'(x;r;a) = Yl^=i ai(x)v(x;r\a,y+1 and letting t = v(x;r;a), we get the required identity of power series. From the identity indicated above, we can also find a formal solution of (3.1) corresponding to a G X of the form (3.10) where

Vi are polynomials in u\,..., ut, v\,..., f,_i without constant terms. In particular, each Vi is a 27r-periodic Lipschitz function which vanishes at 0. Since a 6 X, there is also a solution given by (3.10). These solutions coincide. Thus, we get the identity v(0; r; a) = v(2ir; r; a) = r for any sufficiently small r. •

3.3 The Universal Center

In this section we consider the concept of the Universal Center as described in [9]; before we present a definition of this subclass of centers, we will look at operator- valued center problems. We introduce the associative algebra A(Xi,X2) with unit / of non-commutative polynomials with complex coefficients in variables X\, Xt and / satisfying the relations

pi(X1,X2,I) = 0 for i = l,2,...,m,

are where pt e C^i,^,^] holomorphic polynomials. Then ^4(X!,X2)[[i]] is the associative algebra of formal power series in t whose coefficients are elements from 1 A(Xi,X2). For some a — (ai,a2,...) £ X, we consider the equation on S (the unit circle)

HF °° i 1 — = C£ai(x)t XlXt )F. (3.11) i—l

This equation can be solved using Picard iteration to obtain the local solution F with values in the group (?(X1,X2)[[£]] of invertible elements of «4(Xi,X2)[[i]] whose 43 coefficients are Lipschitz. The monodromy of (3.11) is a homomorphism

p:Z^G(XuX2)[[t}} where Z = 7Ti(51) is the fundamental group of S1. We can use the monodromy as a topological tool to determine when a center exists. When the monodromy is trivial, all elements of the fundamental group of Z are taken to the unit / of G(Xi, X2)[[t]]. This relationship determines a center for (3.11). We denote the center set of (3.11) by

CA = {aeX:p(7r1(Z)) = I}.

Theorem 3.3.1. Suppose A(X1,X2) is determined by the (unique) relation

[Xi,X2] := XiX2 — X2X\ = —X2.

Then C = CA.

Here, we do not provide a complete proof, but give the reader an idea of how this is constructed. Using the definitions from (3.6), we can evaluate elements of

A(Xx,X2) at Xi = D and X2 = L. We verify that these hold by checking the identity DL — LD = L2 clearly, it suffices to check it on elements zk, k > 2.

We have DL{zk) = ^(z^1) = (k - l)zk~2, LD{zk) = L^'1) = kzk~2.

Thus

DL(zk) - LD(zk) = -zk~2 = -L2zk.

2 So DL — LD = —L and evaluation of elements of A(XlyX2) at Xi = D,X2 = L 44

determine a homomorphism of algebra A(X\,X2) onto the algebra A(D,L). In fact it is proved that A(Xi,X2) is isomorphic to A(D,L). But the center problem for ,4(£>, £/)[[£]] is equivalent to the center problem (3.1) as it was established in the proof of Theorem 3.2.4. This gives the required result. Now that we have shown how to determine centers using the operator algebra, we can move onto describing universal centers. The object of this study is to somehow relate the centers from equation (3.1) to (3.11). To do this, we consider another algebra A(F\, F2) with unit / of complex non-commutative polynomials in / and free non-commutative variables

Fi and F2. There are no relations between i<\ and F2. As before ^4(F!,F2)[[t]] is the associative algebra of formal power series in t with coefficients from A(Fi,F2). We consider that there is a surjective homomorphism (f>: ^4(Fi,F2)[[i]] —> ,A(-Xi,X2) [[£]]. We consider for a € X the equation on the circle S1

^ = (^a^mFt1) F. (3.12)

and let p : Z —> G(Fi, F2)[[t\] be the monodromy of (3.12). Then it is clear that

p = 4>op (3.13)

We now have the following Corollary 3.3.2. The center set of equation (3.12) is a subset ofC.

The set U of universal centers for equation (3.1) consists of all elements a 6 X for which the monodromy of (3.12) is trivial. In general U ^ C. Universal centers can also be described in terms of iterated integrals. We start by

% l defining u(x) := X^Si o,i(x)t FiF2~ . We identify the functions on the circle, at(x), with 27r-periodic functions over R. The fundamental solution of (3.12) is a map

F : R -» G(FUF2)[[t]} such that F(0) = I and F'(x) = u{x) • F(x). Using Picard 45 iteration, we can write

F(x):=I + J2 I ••• I u}(sk)---uj(s1)dsk---ds1 (3.14) which can also be written as

oo i F(x) = ^2fi(x;F1,F2)t »=o where fo = I and the other ft are homogeneous polynomials of degree i in F\ and F2 whose coefficients are locally Lipschitz functions in x S R. Then the monodromy of (3.12) is defined as

p := F(2irn) = F(2ir)n, where n e Z.

A known result regarding universal centers is Proposition 3.3.3. Equation (3.1) determines a universal center

(i.e., F(2ir) = p(l) = I) if and only if for all positive integers ii:... ,ik and k > 1

/•••/ aik(sk)---ail(s1)dsk---ds1 = 0. (3.15) J J0

In particular, then

PX di(x) := / a,i(s)ds, z = 1,2,..., Jo are 2-n-periodic Lipschitz functions.

Next, we consider the following finite series of equations

n 1 F'(x) = un(x) • F(x) with un := ^ai(x)f F^' - (3.16) i=l 46

We denote the monodromy of equation (3.16) by p~n : Z —> G(Fi, F2)[[t]]. Proposition 3.3.4. Equation (3.1) determines a universal center if and only if all pn are trivial. Moreover the triviality of pn is equivalent to the fulfillment of equations

(3.15) for any integer 1 < ii:..., i^ < and any k.

Further, the triviality of pn is equivalent to the equation

i+1 fx=£^) (3-17) to determine a universal center. To give a topological description of universal centers we require Definition 3.3.5. The polynomial convex hull K of a compact set if C C is the set of points zgC" such that if p is any holomorphic polynomial in n variables

\p(z)\

A useful corollary which will not be proven here is as follows:

1 n We introduce the Lipschitz map An : S —> C , defined by An(x) = (di(x),..., an(x)), 1 and set Fn := A^S ), Assume that either

1. Tn C X where X is a closed one-dimensional complex analytic subset of a domain f/cC" such that U = (J Kj with Kj C -Kj+i and Kj = Kj for any j, or

n 2. Tn C X where X C C is a connected one-dimensional complex space with dim

Corollary 3.3.6. Let Fn C X where X satisfies (1) or (2). Suppose that there 1 is a continuous map An : R —> X of an open neighborhood flcC of S such that 1 An\si = An, and A~ (x) is finite for any x G An(R). Then the corresponding equation l (3.17) determines a universal center if and only if An = AinoA2n where A2n '• S —-> ID) 47 is a continuous map onto the unit disk DcC, locally Lipschitz outside a finite set, and Ain : ID —> C" is a finite holomorphic map.

Corollary 3.3.7. Suppose that the coefficients <2i,... ,an in (3.17) are trigonometric polynomials. Then (3.17) determines a universal center if and only if there is a trigonometric polynomial q and polynomials pi,... ,pn £ C[z] such that

a,i(x) = Pi(q(x)), x G S1, 1 < i < n. where a,i(x) is as defined above.

Proof. Let C* := C — {0}. If the coefficients a\,... ,an in (3.16) are trigonometric 1 n polynomials, the map An : S —> C can be extended to a holomorphic map An : n C* —> C whose components are Laurent polynomials. If all components of An are either polynomials in z or in ^ the required factorization of An trivially exists. Thus we may assume without loss of generality that at least one component of An is a polynomial in both z and j. Let CPn denote the complex projective space. Then Cn = CPn — H where H is the hyperplane at infinity. Let X be the Zariski closure of

n X = ^4n(C*). Then X is a (possibly singular) rational curve and X = X — H C C is a closed algebraic subvariety. If the monodromy of (3.16), pn is trivial, (3.3.6) implies 1 1 that An : S —> X is contractible. Let v : CP —> X be the normalization of X and X = v~1(X) C CP1 be the normalization of X. Then the following commutative diagram is valid:

X C ^CP1 v X X {c •^ Xv

We also have a lift which gives us Ain : C* —• X demonstrated by the diagram

X A, * / / c*—^^x

such that An = v o A\n. Now, clS -fiyi IS contractible and v is the normalization of

X, Ain is also contractible. By the construction of A\n, we know it is a finite proper surjective map; thus the image of the induced homomorphism of fundamental groups

1 (iln). : MS ) - 7n(X)

is a subgroup of finite index in TTI(X). NOW as Ain is both surjective and contractible, 1 n m{X) = {1}. Thus X^CcCP . Since v : X -f C and iln are both algebraic, there are polynomials Pi,... ,pn G €[2;] and a Laurent polynomial g such that i/(z) =

(^1(2;),... ,pn(z)) for ^ G C, and Ai„ = q(z) for 2; € C*. Thus, Oi(x) = Pi{q{x)) for x G 51 and 1 < i < n. Conversely, we need to show that the above factorization implies that the monodromy of (3.16) is trivial. Now, using An as above, we can pull back the equation (3.16) so that it acts on C* instead of on the space of 2TT periodic functions over M. Thus,

u)n = uno An

We start with recognizing the induced homomorphism (An)* : 7Ti(C*) —> iri(X).

Also, we note that p is the monodromy for the equation dF = ujn • F, and p is the monodromy for dF = un • F which gives us the equation

p = p°(4)*- 49

We now perform a lift on the diagram below to define A'n.

X A' S / V / C*^^ X An

Thus, we now have

LOn = Cunopo A'n.

Hence, there are two induced homomorphisms p* : TTI(X) —• iti(X), (A^)* : 7Ti(C*) —> 71"! (X). so that the monodromy of (3.16) splits into

p = pop,o(A'n),. (3.18)

If we consider An to be acting from CP to CP™, we can use homogeneous coordinates to compute where 0 and oo are taken to be in the image space. We see that under all nontrivial Laurent polynomial maps 0 and oo are taken to oo and under all polynomial

n maps oo is taken to oo. Thus, An(0) = An(oo). Consider ^4„(CP ) to be covered by the space CP via the map p; then, if we restrict the domain of An to C, we find 1 that there is only one point p~ (An(0)) which must be subtracted from CP in order for the diagram to commute. Thus, X = CP - {p-^i^O))} = C. Therefore the homomorphism (A£J* is trivial, and (3.18) is trivial. •

Corollary 3.3.8. Suppose that the coefficients a^,..., an in (3.16) are complex poly nomials. Then (3.16) considered on [a,b] determines a universal center if and only if there are polynomials q,pi,---,pn £ C[z] such that

q(a) — q(b) and hi = Pi(q), 1 < i < n. 50

Proof. If (3.17) determines a universal center, than as in (3.3.3) all iterated integrals from ai,...,an are zeros. Then in particular, a,i(a) = Sj(6) = 0Vi. If we identify the l unit circle S with M/((6—a)Z) we can think of v4„(X) := (&i(:c),... ,an(x)),x & [a,b], 1 as a Lipschitz map S —• C™. Substituting to the last formula x S C, we extend An n to a holomorphic polynomial map An : C —* C . Then X = ^4n(C) is a (possibly singular) rational curve. Since (3.17) determines a universal center, by (3.3.6), AN : Sl —> X is contractible. Additionally, let ^ : C —• X is the normalization map of

X. Thus, there is a polynomial q € C[z] such that An = u o q. Since i/ is algebraic we write u(z) := (pi(z),... ,pn(z)),z € C, for some Pi,... ,pn G C[,z]. Suppose that g(a) 7^ g(6). Since ^4„(a) = An(b), the normalization sews together q(a) and X is not contractible. Thus q(a) = q{b). Conversely, suppose that a^ = p^o q and q,pi G C[z] for 1 < i < n, and

can 11 q(a) = q(b). Then q\\a,b] be considered as a map from S —+ C. We can also write

An = v o g| [„_&]. Thus, as in the proof of (3.3.7), we see that the following diagram that commutes is valid.

^-7— X

We show that the monodromy is trivial in a similar manner. Let p be the monodromy for the differential equation (3.16), and let p be the monodromy for differential equa tion produced by dF = A*nu)nF. Then, we find that

where v* and g* are induced homomorphisms on their respective fundamental groups. Now as g* = 1*, and p o v* must preserve identity, p — 1*, a universal center is 51 determined. •

Corollary 3.3.9. Let now H{x, y) G C[x, y] be a homogeneous polynomial. For any holomorphic functions P\,P2, defined in an open neighborhood of 0 G C we define A(x,y) := Pi(H(x,y)), and B(x,y) := P2{H(x,y)). Then the vector field

dA{ x y) 2dA y) x = -y- xy d x' + x ^ - yB(x, y)

2 y = x- y ^) + xy™j?fl + xB(x,y) determines a center.

Proof. Passing the equation to polar coordinates, we obtain the equations

dr P d(f> 1 + Q where

P(r,cf>) = ^^ and Q(r, 0) = B(r, 0)

We also write H(r, 4>) = h((f>)rk in polar coordinates where h is a trigonometric polynomial of degree k. This allows us to write P and Q as series.

ki d<\>

b ki Q(r,(f>) = f2 kti()r i=0 52

The differential equation above can be expressed as

dr _ZZiaihi~1()h'()rki

= *'(*): oo

= $>(W)) •&'(*)»•'• i=l for small enough r and pi G C[z\. The first integrals of these coefficients are

o*= / Pi(h())h'(4i)d(f> Jo = Pi(h{x)) where Pi(x) = \ Pi(s)ds. Jo

Thus, the first integrals of the coefficients of the system are as those described in Corollary (3.3.7), thus, a universal center is determined. •

Corollary 3.3.10. Let F(x,y),G(X,Y) be analytic functions defined in an open neighborhood o/O £ R2 such that their Taylor expansions at 0 do not contain constant and linear terms. Suppose F(x, —y) = —F(x,y) and G(x, —y) = G(x,y). Then the vector field

x = -y + F(x,y)

y = x + G(x,y)

determines a center.

Proof. We begin by putting the system into polar coorindates as in section 2.3. We find that

dr F(rcos6),rsm6)cos0 + G(r cos 6, r sin 9) sin 0 d6 r — F(r cos 6, r sin 6)r sin 6 + G(r cos 9, r sin 9) cos 9 53

Thus if we write

P(r, 9) = F(r cos 9, r sin 9) cos 9 + G(r cos 9, r sin 9) sin 9

Q(r, 9) = r — 1 — F{r cos 9, r sin 6) sin # + G(r cos #, r sin 0) cos # we can verify that P is odd and Q is even in 9. We show, e.g., that P is odd.

P(r,-0) = F(rcos(-9),rsm(-9))cos(-9) + G(rcos(-9),rsm(-9))sm(-9)

= F(r cos 0, — r sin 0) cos 9 — G(r cos 0, — r sin 0) sin 9

= —F(r cos 0, r sin 0) cos 0 — G(r cos 9, r sin 9) sin 0

= -P(r,0).

As P is odd and Q is even, we can write

oo oo

P(r,0) = ^2Pj(9y and Q(r,0) = $>-(0)r>' such that pj G span{sin(n#) :n£N} and qj € span{cos(n#) : n G N}. Thus, if we take the integral of pj, we find

pj(9) = /f" Pj(s) ds € span{cos(ra#) : n € N} Jo

Therefore as in Corollary 3.3.9, if we let H(r, 9) = cos# we can use the same method from Corollary 3.3.9 to show that the first integrals of the coefficients satisfy the hypothesis of Corollary 3.3.7 as follows: let

, , df°P(r,e)dr , N , x S(r,9) = Jo ^ ' > and T(r,9) = Q(r,9). 54

Then S and Q can be written as a series of the form

oo 1 l S(r, e) = J2ak cos'" 6 sin 6 r and

oo T(r,6) = J2hcosi6ri. i=i

Thus, as in corollary (3.3.9), for small enough r, we can write ^ = Xw^i Pi(cos #)-sin 0; therefore giving us first integrals of the coefficients which satisfy corollary (3.3.7). •

3.4 Systems Generating the Group of Rectangular Paths

In this part, we described the only known to present class of equations (3.1) for which the center problem can be completely solved. First, as in [11], we define some structure on the center set C of equations (3.1) obtained in [10]. Recall that for a = (ai, a2, • • •) € X, the first integral

a(x) := f / a,i(s)ds, / a2(s)ds,. is a path from IT '•= [0,T] into C°°. We consider these paths with the operations o and _1 given by { ~b{2x) if 0 < x < T/2, b(T) - 5(0) + d{2x - T) if T/2 < x < T,

a_1(x) :=a(T-x),0

We now let C {X\, X2,...) be the associative algebra with unit / of complex 55

noncommutative polynomials in / and free noncommutative variables Xx, X2,.. • , i.e., there are no nontrivial relations between these variables. By C (Xi,X2,...) [[£]] we denote the associative algebra of formal power series in t with coefficients from

C (Xi,X2,...}. Also by A C C (XUX2,...) [[i\] we denote the subalgebra of the form

n / = Q,/ + f;( J2 ciu...,ikXh • • • Xik\ t (3.19) n=l \iiH Mfc=n / where Co, Q^...^ € C for alHi,..., ifc, A; € N. We equip A with the weakest topology in which all the coefficients Cilt_,jk in (3.19) considered as functions in / G A are continuous. By G C A, we denote the subset of elements / of the form (3.19) with c0 = 1. Then (G, •) is a topological group. Its Lie Algebra £G C A is the vector space of elements of the form (3.19) with CQ = 0; here for f,g(z £G, their product is defined by the formula [f,g] := / • g — g • f. Also, the map exp : Cg —> G defined by exp(/) := e^ = X^o j, is a homeomorphism. For an element a E X, consider the equation

i F'(x) = l^ai{x)Xit ] F(x), where x e IT. (3.20)

This equation can be solved by Picard iteration to obtain a solution Fa : IT —> G with the initial condition Fa(0) = I where the coefficients in expansion in Xi, X2,..., and t are Lipschitz functions on It. Consider the function E(a) := Fa{T) for a G X. Then one can check that E(a * b) = E(a) • E(b) for a, b € X. Explicitly, we find that

E-.X^G is defined by

n E(a) = / + E ( £ Ih,...,ik{a)Xh ...Xik\ t , 56 where

a Iiu-,ik( )'•= /'•'/ aifc(sfc)---ail(si)rfsfe---rfsi J Jo

are basic iterated integrals on X.

The kernel of the homomorphism E coincides with the set of universal centers U of the equation

oo + ^ = J>(*K \ XG[0,2TT]. (3.21)

of Section 3.3. The set of equivalence classes G(X) := X/ ~ with respect to the equivalence relation

a ~ b 4=> a * b"1 € U

has the structure of a group, so that the factor-map ir : X —• G(X) is an epimorphism of groupoids sending each a-1 € X into (7r(a))_1 € G(X), moreover for each function

on hi,...,ik -X", there exists a function Iilv..,tfc on G(X) such that hi,...,ik ° TT = hi,...,ik- In particular, there exists a monomorphism of group

£ : G(X) -• G defined by

E = EOTT, i.e.,

B £%) = / + £( E 4....,ifc(^ii---^J* , jeG(4

We equip G(X) with the weakest topology in which all functions Iilt...,ik are contin uous. Then G(X) is a topological group, and E is a continuous embedding. The completion of the image E(G(X)) C A is called £/ie group of formal paths in C°° and is denoted Gf(X). 57

Let G[[r]] be the set of formal complex power series f(r) = r + ]CSi difl+1 where di : G[[r]] —> C be such that di(f) is the (i+l)-th coefficient in the series expansion of /. We equip G[[r]] with the weakest topology in which all di are continuous functions and consider the multiplication o in G[[r]\ defined by the composition of series. Then

G[[r]] is a separable topological group. By Gc[[r]] C G[[r]] we denote the subgroup of power series locally convergent near 0 equipped with the induced topology. Next, we define the map

P:X^G[[r]] by

i+ P(a)(r):=r + fM £ Piu...M • 4,,<» ) r \ (3.22) i=l \iiH Hk=i / where Pilt...,ik(t) := {t — i\ + l)(t — i± — i2 + 1)... (t — i + 1). Then one can check that

P(a * b) = P(a) o P(b) and P(X) = Gc[[r]]. Moreover, let v(x;r;a),x G It, be the Lipschitz solution of equation (3.21) with initial value u(0;r;a) = r. Then for every x G It, we have v(x; r;a) & Gc[[r]]. It is shown in Section 3.2 that P(a) = v(T; .;a). In other words, P(a) is the first return map of (3.21). Given this fact, we have it that

i a fora11 i e N a6C^=> ^ Pii,...,

Using (3.22), we can create a continuous group homomorphism

P : G(X) -y G[[r]] defined by

P = P0TT.

Identifying G(X) with its image under E, we extend P by continuity to G/(X) 58 obtaining the map P for the extension

G[ Ml

G(X)- -»- G + GF(X) E completion

Without loss of generality, we may assume that P is an extension of P so that in what follows, we may use the symbol P instead of P for this map. We set C := ir(C) and define Cf as the completion of E[C). Then Cf coincides with the kernel of the homomorphism P. The groups C and Cf are called the group of centers and formal centers of equation (3.21). It has been established in [9] and [10] that

geCf4=^ Yl Pii,...,iJii,..,ik(9) = 0 for all i € N. h-\ Hk=i

fora11 Yl Pii,...,tfc(*)-4,...,ifc(3) = 0 »6N. »iH l-*fc=*

3.4.1 Group of Piecewise Linear Paths

The main topic of this section is the group of rectangular paths; however, the subject naturally extends to the group of piecewise linear paths. There is more work available in this group; however, progress is blocked by some hard conjectures which need to be proved. In this subsection, we will introduce this group and then carry on to the main topic. Consider the elements g G Gf(X) of the form

h = e , where /I^PQX^, c^eC^eN. (3.23) i=l 59

By PL C Gf(X) we denote the group generated by all such g. It is called the group of piecewise linear paths in C°°. The first integrals of the vector of coefficients of the formal equation (3.21) corresponding to elements of PL are piecewise linear paths in C°°. It has been shown in [10] that the group Cpi := PL D Cf of piecewise linear centers is dense in Cf. In [10] it was asked about the structure of the set of centers represented by piecewise linear paths in C™. In the following subsection, the solution of this problem is presented for the group of rectangular paths.

3.4.2 Center Problem for the Group of Rectangular Paths

Consider the center problem for

, k ^ = J>(*K+\ x£lT, (3.24) i=i with the following conditions. Let Xrect C X be the subset generated by elements a E X such that the first integrals a are rectangular paths in C°°. More specifically, these paths are composed of finitely may segments parallel to the coordinate axes.

Each a E Xrect is the vector of coefficients of equation (3.24) where k = k(a) E N and a, are defined as follows. There exists a subdivision of the interval IT into k closed subsets Ji,..., Jfc such that each J, is the union of finitely many nontrivial closed subintervals of IT and each Ji D Jm is either 0 or consists of finitely many points.

Then a* is a function from L°°(IT) equal to 0 on IT\JI- The image of Xrect under the homomorphism E is called the group of rectangular paths and is denoted by G(Xrect). 0 Clearly G(Xrect) C PL. It is generated by elements e "*"*" E Gf(X),cn E C,n E N 1 Cr,Xntn as the preimage £'~ (e ),cn E C consists of elements an E Xrect of the form

&n = («i, 0-2, • • •), where a^ = 0 for i ^ n and the function an E L°°(IT) is such that

T L an(t) dt = cn. 60

We see that any rectangular path modulo a Lipschitz reparameterization is the com position of the paths created by the first integrals an of the elements an for n € N. CnXntn Since there are non nontrivial relations between elements e with cn ^ 0, the group G(Xrect) is isomorphic to the free product of countably many copies of C. Also, the group G(Xrect) C G(X) is dense in Gf(X). For each g € G(Xrect) the first return map P(g) € Gc[[r]] can be explicitly computed and represents an algebraic function. In fact, for the equation

dv cn Tx = T'Vn+1

CnXntn corresponding to the element gn := e , an explicit calculation shows that its first return map is given by the formula

P(9n){r): = yi -„V" = r+g:(-iyw-i)+i)(no-2)+i)-i4|J,+li (325)

3=1 J'

Here, we define V7 : C\(—oo, 0] —> C as the principal branch of the power function.

Then, for a generic g G G(Xrect), the first return map P{g) is the composition of series of the form (3.25).

Theorem 3.4.1. The restriction P\G(xrect) '• G(Xrect) —> Gc[[r]] is a monomorphism.

In particular, C D G(Xrect) = {1} and C D Xrect C U.

Proof. The proof of this theorem is based on a deep result of S. Cohen [13]. Consider

Ck Xk t an irreducible word g = g^ • • • g^ G G(Xrect), where g^ := e i i * and c^ € C* := C\{0}. We must show that P(g) = P{gki) ° • • • ° P(dki) ¥" lj here 1 is the unit of the group Gc[[r]]. Assume on the contrary that P(g) = 1. Then from equation (3.25), for 61 all r G C sufficiently close to 0 we get

P(9)(r) = -. (3.26) ( kl-2 fe kl-l fe k fc 1 kl (l + 4,r <) i + 4!_1r <~ + + dklr \

Here, we have set dki := —kiCki. We now make a substitution t = ^ to get from equations P(g) = 1 and from (3.26), we get for all sufficiently large positive t,

( kl-2 fc N \fe \ f-l \ kl-l kl kl \ {t +dkl) +d _ + (3.27) kl 1 ) ^ 1 +4i = t. \ /

From the irreducibility of g, we find that ki ^ fa+i, for all 1 • C, and a (single valued) holomorphic function / defined on S, such that the pullback by 7r_1 of the restriction of / to a suitable open subset of S corresponds to the branch of the original function satisfying (3.27), defined on an open subset of C containing a ray [R, oo) for R sufficiently large. Equation (3.27) implies that / coincides with the pullback ir*z of the function z on C.

Let P be the Abelian group of maps C —> C generated by {x Hip;pg N}. Here the inverse of the complex map x >—> xp is the map x i—> x1^, which takes a nonzero x into the unique complex number y with 0 < argy < —, such that yp = x. Let Tc be the Abelian group of maps {x \—• x + a; a G C}, then [13] asserts that the group of complex maps of C generated by P and Tc is their free product P * Tc-

Now, the function h : C —• C defined by the left-hand side of (3.27) belongs to the group generated by P and Tc (in the definition of h we define the fractional powers as in the above cited theorem). In turn, by the definition of S there exists a subset 62 u of S such that IT : U —> C is a bijection and / o (7r|{/)_1 = h. Since / = 7r*2, the latter implies that h(t) = t\/t E C. But according to our assumptions, the word in P * Tc representing h is irreducible. Thus it cannot be equal to the unit element of the group. The contradiction shows that P(g) ^ 1 and proves the first statement of the theorem. The second statement follows straightforwardly from the corresponding definition. •

Combining some results of [9] with Theorem 3.4.1 we get the description of centers of equations (3.24) satisfying the corresponding condition. To formulate our further result, we introduce some more notation. Consider the Lipschitz curve

k a(x) := ( / ai(«)d«,..., / ak(t)dt) : IT -+ C

Let Xu be defined as either case

fc (a) If O,{IT)

fc X = \Jj=1Lj of complex lines Lj C C each parallel to one of the coordinate axes fc in C . The universal covering p : Xu —> X of X is a contractible one-dimensional

complex analytic space. It is the union of subsets Ljg C Xu, 1 < j < s, g E ni{X)

such that P\LJ9 maps Ljg biholomorphically onto Lj,l < j < s,g E TTI(X). fc (b) If a(IT) C M , that is all a» are real valued, then O,(IT) belongs to the union s k X = U j=1Lj of closed intervals Lj C M. each parallel to one of the coordinate fc axes in R . The universal covering p : Xu —> X of X is a metric tree with the set

of edges Ljg, 1 < j < s,g E iri(X), such that p\ijg maps Ljg bi-Lipschitzly onto

Lj,l < j < s,g E TTI(X). We equip X with the Riemannian structure induced from Rk. Theorem 3.4.2. Suppose that coefficients ai,...,a,k of equation (3.24) satisfy the associated condition given in (3.24). This equation determines a center if and only

1 if there exists a Lipschitz map A : IT —> Xu such that the preimage A~ {Ljg) of each 63

Ljg is either 0 or is the union of finitely many points and closed subintervals of IT, i(0) = A(T) anda = poA.

Proof. Suppose that equation (3.24) with the associated condition and coefficients ai,..., % determines a center. According to Theorem 3.4.1 this center belongs to the set of U of universal centers. From [9], Theorem 1.10 as in the proof of [9], Corollary

1.12, we obtain that since 5,(IT) belongs to a triangulizable space X, there exists a continuous map A : IT —> Xu such that A(0) = A(T) and a = p o A. Since by the associated condition, the preimage A~l{Lj) of each Lj is either 0 or is the union of finitely many points and closed subintervals of IT, from the identity A = p o A, and 1 using the structure of Xu, we obtain that the preimage A~ (Ljg) of each Ljg is either 0 or is the union of finitely many points and closed subintervals of IT as well. Moreover, since each Ljg is bi-Lipschitz equivalent by means of p to Lj and a is a Lipschitz map, from the identity a = poAwe obtain that A is Lipschitz too. Suppose now that under the above notation there exists a map A : IT —> Xu satisfying the conditions of the theorem. Let us show that the corresponding equation (3.24) determines a universal center. Consider equation (3.20) corresponding to the equation (3.24)

i F\x) = I J2 ai{x)Xit j F(x), x e IT. (3.28)

Let pi,...,pk be coordinates of the map p. Since Xu is Lipschitz triangulizable and k the map p : Xu —>• C is Lipschitz, we can define differential forms dpi,..., dpk almost everywhere on Xu (as pullbacks with respect to p of differential forms dz\,..., dzk on fc fc C ; here Z\,...,Zk are complex coordinates on C ). We set UJ :— JZi=1 Xftdpi and consider equation dG = UJG on Xu. By definition, u is the differential of a Lipschitz function on Xu with values in the algebra C (X±,..., Xk) [[t]]. Also, dco = w Aw = 0

(because Xu has complex or real dimension 1), and Xu is contractible. Thus, using

Picard iteration on each Ljg we can globally solve equation dG = uiG to obtain a 64

solution G : Xu —> C (Xi,... ,Xk) [[t]] whose coefficients in the series expansion in

X\,... ,Xk and t are Lipschitz functions on Xu and such that G(^4(0)) — I. Next, from the identity a = p o A we obtain that F := G o A, the fundamental solution of equation (3.28), determines a universal center. •

Consider the Abel differential equation

2 3 v' = (n(x)v + a2(x)v (3.29)

with real-valued continuous functions ai, a2. In case a\, a2 are polynomials, the major conjecture asserts that this equation determines a center if and only if

Oi{x) = A\{P{x)) • P'{x),x ElT,i = 1,2, where Ai and P are real polynomials and P(0) = P(T). This condition is similar to the condition of Theorem 3.4.2. The conjecture is checked to be true for several classes of Abel differential equations with polynomial coefficients; see [36], [28]. It is known that an analogous statement is wrong for centers of Abel differential equations whose coefficients are trigonometric polynomials; see [27].

3.4.3 Bautin Problem for the group of rectangular paths

In this section, we present a solution of the Bautin type problem for the group of rectangular paths.

Let a € Xrect be such that

1 Ck Xk tk E(7r(a)) = e^i**!** • • • e i i ' e Gf(X) for some c^,..., c^ € C, i.e, the path a : IT —> C°° consists of / segments parallel to the coordinates axes ^,...,2^ of C°°. Let sx < s2 < • • • < sm be the set of 65 distinct numbers among ki,...,k\ and Sj :— {i G {ki,... ,ki} : U = Sj}. Considering Cfcj,..., Cfc, as complex variables in C', we obtain a family T of rectangular paths depending holomorphically on a parameter in C'. Elements a G Xrect corresponding to paths of this family are vectors of coefficients of equations (3.24) satisfying the associated condition of the form

(ill s +1 Sm+1 — = aSl(x)v ' + ••• + aSm{x)v ,x G IT, (3.30)

where aSj is an L°° function on IT equal to 0 outside a family {Ii}ieSj of mutually disjoint open intervals and such that JL aSj(t)dt — Cj. The first return maps P{a) of elements of T can be computed by expanding the functions eCfciXfci' 1 • • • eCkiXkit l hs ks x in infinite series in variables Xkst , 1 < xs < I, then replacing each Xk3 by DL ~ where D and L are the differentiation and the left translation in the algebra of formal power series C[[z]], and then evaluating the resulting series in D,L,t at elements zv\ see [9] for similar arguments. As a result we obtain (with r in place of t)

00 ( cSl cSl\ P(a)(r) = r + £ f £ ftx;*....*:.,^ • • • j\) ri+1 (3.31) i=l \fcisiH \-kisi—i ' ' / i l-l (sn+l / n Smkm kn+i+i\^\ 3 32 9feii8i,...,fc,;«,() = n i n (* ~ YI ~ i )\ ^- ^ n=0 L j=0 \ m=0 / J

(we set s0^o := 0 for convenience).

By Pi(ck!, • • •, Cfe,) we denote the coefficient at rt+1 of P(a). It is a holomorphic polynomial on Cl. The center set C? of equations (3.30) corresponding to the family T is the intersection of sets of zeros

{(Cfei.---.Cfc,) GC' :pi(ckl,...,Ckl) = 0}

of all polynomials pt. According to Theorem 3.4.1, (c^,..., CfeJ G C if and only if 66 the word eCfci* 1Xfci • • • e0^ iXki — I G A. Since the groups generated by elements eCfc'i fc(i,..., eCfcim m fc'm with mutually distinct .Xfc, and nonzero numbers c^, are free, the last equation implies that this center set C is the union of finitely many complex subspaces of C' (for instance, if all kj are mutually distinct, then the center set corresponding to the family defined by (3.30) is {0} C Cl). The next result gives an effective bound on the number of periodic solutions (i.e., solutions v for which v(0) = v(T)) with sufficiently small initial values for equations (3.30) not determining centers (solving the Bautin problem in this case; see [18]). This also gives a bound on the number of coefficients in (3.32) determining the center set C? Theorem 3.4.3. Let B C Cl be the open Euclidean ball. Consider the family of equations (3.30) with (c^,..., c^,) G B. Then there exists a positive number R such that the number of periodic solutions v with initial values satisfying \v(0)\ < R for equations of this family not determining centers is bounded by

d •= TT kk

Moreover, the set Cjr c C is determined by equations p\ = 0,... ,Pd+i = 0.

kl l Proof Since p^z^c^, • • •, z Ckt) = z pi(ck1,..., Ck,), i G N, it suffices to prove that Cj: Pi B is determined by equations p\ = 0,... ,pd+i = 0. Next, there exists a positive number R such that for each (c^,..., c^) G B so lutions v with initial values satisfying |u(0)| < R of the corresponding equations

(3.30) with vectors of coefficients a G Xrect are well defined on IT and the corre sponding first return maps P(a)(z) given by (3.32) are holomorphic functions on the disk 3ft := {z G C : \z\ < R}. On the other hand, according to (3.25), P(a) is the composite of algebraic functions. Now from formula (3.26) we obtain (estimating the degree of the graph in CP2 of the multi-valued algebraic function defined by the

a z z denominator of (3.26)) that equation f(a)(z) = c, f(a)(z) := ( >( '~ : has at most d 67 complex roots in BR (counted with their multiplicities), i.e., the valency of /(a) on H>R is at most d. This gives the required bound on the number of periodic solutions v with \v(0)| < R for equation (3.32) corresponding to a. Also from the estimate for the valency by the result of Haymann [22], Theorem 2.3, we obtain

1+ 2d Rd+1 — 1 sup \f{a){z)\ < e 2 .——— .max{l

This implies that if pi(ckl,..., ckl) = •••= pd+1(ckl,..., ckl) = 0, then f(a) = 0,i.e.,(ckl,...,ckl)eC. D

It follows from equations

kl kl l Pi{z ckl,...,z ckl) = z pi(ckl,...,ckl), i GN, inequality (3.33), and the Cauchy integral formula for derivatives of /(a), that for any k € N and all A G Cl,

|PWA)| < e » • ^^-p • ^—j • (1 + ||A||a)^ • i max+i |Pfc(A)|,

where || • ||2 is the Euclidean norm on C'. Similar inequalities for Taylor coefficients of families of univariate holomorphic functions depending analytically on a parameter originally appeared in [18] and were applied to problems of bifurcations of periodic orbits of differential equations such as the local version of Hilbert's 16th problem. The above inequalities and [21] Proposition 1.1, imply that each polynomial pa+i+k belongs to the integral closure I of the polynomial ideal / generated by pi,... ,Pd+i- 68

Chapter 4

Further Work

The field of planar polynomial vector fields is rich with potential. Although many years without results passed, the new algebraic approach has created a new vessel towards results. At this point in time, the author does not intend to pursue a Ph.D; however, it is useful to discuss what kinds of further work would be attempted and to where this further work could lead. Section 3.4.1, Group of Piecewise Linear Paths, was introduced without any results regarding the piecewise linear paths. The purpose of the section was to discuss the general terms used to talk about this group. In, 3.4.2 Center Problem for the Group of Rectangular Paths, we restricted our investigation of the group of piecewise linear paths and their relation to the center problem to those paths which are parellel to an exis in X. There is a strong intuition to believe that there is a similar result to Theorem 3.4.3 for not just recangular paths but for piecewise linear paths as well. Considerable effort would be needed for a positive result. In addition to the aformentioned, results similar to Corollaries 3.3.9 and 3.3.10 may be discovered by studying different classes of polynomial equations. 69

Bibliography

[1] M. A. M. Alwash, On the composition conjectures, Electronic J. of Diff. Equations 69 (2003), 1-4. [2] M. A. M. Alwash and N. G. Lloyd, Non-autonomous equations related to poly nomial two-dimensional systems, Proceedings of the Royal Society of Edinburgh 105A (1987), 129 - 152. [3] V. I. Arnold, Developments in mathematics: The Moscow school, ch. Problems on Singularities and dynamical systems, pp. 251-274, Chapman & Hall, London, 1993. [4] V. I. Arnold and Yu. Il'yashenko, Ordinary differential equations, encyclopedia of mathematical sciences, Springer, 1988. [5] N. N. Bautin, On the number of limit cycles which appear with the variation of coefficients from an equilibrium point of focus or center type, Translations Amer. Math. Soc. 1 (1962), 396-413. [6] I. Bendixon, Sur les courbes defines par des equations differentielles, Acta Math. 24 (1901), 1-88. [7] M. Blinov, Center and composition conditions for Abel equation, Ph.D. thesis, The Weizmann Institute of Science, Rehovot 76100, Israel, 2002. [8] A. Brudnyi, An explicit expresion for the first return map in the center problem, Journal of Differential Equations 206 (2004), 306-314. [9] , On the center problem for ordinary differential equations, American Journal of Mathematics 128, no. 2 (2006), 419 - 451. [10] , Formal paths, iterated integrals and the center problem for ordinary differential equations, Bull. Sci. Math 132 (2008), 455-485. [11] , Center problem for odes with coefficients generating the group of rect angular paths, C. R. Math. Rep. Acad. Sci. 31 (2009), 33 - 44. 70

C. Christopher, An algebraic approach to the classiftction of centers in polynomial Lienard systems, J. Math. Ann. Appl 229 (1999), 319-329. S. D. Cohen, The group of translations and positive rational powers is free, Quart. J. Math. Oxford 46 (1995), 21 - 93. C. B. Collins, Conditions for a center in a simple class of cubic systems, Diff. and Int. Equations 10 (1997), 333-356. J. Guckenheimer D. Schlomiuk and R. Rand, Integrability of plane quadratic vector fields, Expositiones Mathematicae 8 (1990), 3-25. G. Darboux, Memoire sur les equations differentielles algebrique du premier ordre et du premier degre (melanges), Bull. Sci. Math. Ser. 2 2 (1878), 60 - 96; 123-144; 155-200. J. P. Frangoise, The successive derivatives of the period function of a plane vector field, J. Differential Equations 146 (1998), 320 - 335. J.-P. Francoise and Y. Yomdin, Bernstein inequalities and applicaitons to anlytic geometry and differential equations, J. Funct. Anal. 146 (1997), 185-205. M. Frommer, Die integralkurven einer geohnlichen differential-gleichung erster ordnung in der umgebung rationaler unbestimmtheitsstellen, Math. Ann. 99 (1928), 222-272. V. A. Gaiko, Global bifurcation theory and Hilbert's sixteenth problem, Kluwer Academic Publishers, 2003. J.-J. Risler H. Hauser and B. Teissier, The reduced Bautin index of planar vector fields, Duke. Math. J. 100 (1999), 425-445. W.K. Hayman, Multivalent functions, second edition, Cambridge Tracts in Math ematics 110 (1994). Seok Hur, Composition conditions and center problem, CRAS Paris 333, Ser. I (2001), 779 - 784. Yu. Il'yashenko, Algebraic unsolvability and almost algebraic solvability of the 71

problem for the center-focus, Funct. Anal. Priloz 6 (1972), No, 3, 30-37. [25] Yu. Ilyashenko, Centennial history of Hubert's 16th problem, Bulletin of the American Mathematical Society 39,no. 3 (2002), 301 - 354. [26] A. M. Lyapunov, The general problem of the stability of motion, Translated by A.T. Fuller from Eduard Darboux's French translation (1907) of the 1892 Russian original, Internat. J. Control 3 (1992), 521 - 790. [27] J.P. Francoise M. Briskin and Y. Yomdin, The Bautin ideal of the Abel equation, Nonlinearity 11 (1998), 431-443. [28] M. Muzychuk and F. Pakovich, Solution of the polynomial moment problem, http://arXiv:0710.4085. [29] D. O'Shea, The Poincare conjecture, Walker & Company, 2007. [30] H. Poincare, Memoire sur les courbes definies par les equations differentielles, J. de Mathematiques 7 (1881), 375-422. [31] , Memoire sur les courbes definies par les equations differentielles, J. Math. Pures Appl 1 (1885), 167-244. [32] H. Poincare, Sur les courbes definies par une equation differentielle, Oeuvres t.l (1892). [33] D. Schlomiuk, Algebraic particular integrals, integrability and the problem of the center, Trans. AMS 338 (1993), No. 2, 799 - 841. [34] , Aspects of planar polynomial vector fields: global versus local, real versus complex, analytic versus algebraic and geometric in normal forms, bifurcations and finiteness problems in differential equations, Kluwer Academic Publishers, Netherlands, 2004. [35] K. S. Sibirsky, Nonlinear science: Theory and applications, ch. Introduction to the Algebraic Theory of invarients of differential equations, Manchester Univer sity Press, Manchester, 1988. [36] Y. Yomdin, The center porblem for the Abel equation, compositions of functions 72

and moment conditions, Moscow Math. J. 3 (2003), 1167-1195. [37] H. Zoladek, The problem of center for resonant singular points of polynomial vector fields, J. Diff. Equations 137 (1997), No. 1, 94 - 118.