On the Center Problem for Ordinary Differential Equations
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UNIVERSITY OF CALGARY On the Center Problem for Ordinary Differential Equations by 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 © Douglas McLean 2010 Library and Archives Bibliotheque et 1*1 Canada Archives Canada Published Heritage Direction du Branch Patrimoine de I'edition 395 Wellington Street 395, rue Wellington Ottawa ON K1A 0N4 Ottawa ON K1A 0N4 Canada Canada Your file Votre reference ISBN: 978-0-494-69427-5 Our file Notre reference ISBN: 978-0-494-69427-5 NOTICE: AVIS: The author has granted a non L'auteur a accorde une licence non exclusive exclusive license allowing Library and permettant a la Bibliotheque et Archives Archives Canada to reproduce, Canada de reproduce, publier, archiver, publish, archive, preserve, conserve, sauvegarder, conserver, transmettre au public communicate to the public by par telecommunication ou par I'lnternet, preter, telecommunication or on the Internet, distribuer et vendre des theses partout dans le loan, distribute and sell theses monde, a des fins commerciales ou autres, sur worldwide, for commercial or non support microforme, papier, electronique et/ou commercial purposes, in microform, autres formats. paper, electronic and/or any other formats. The author retains copyright L'auteur conserve la propriete du droit d'auteur ownership and moral rights in this et des droits moraux qui protege cette these. Ni thesis. Neither the thesis nor la these ni des extraits substantiels de celle-ci substantial extracts from it may be ne doivent etre imprimes ou autrement printed or otherwise reproduced reproduits sans son autorisation. without the author's permission. In compliance with the Canadian Conformement a la loi canadienne sur la Privacy Act some supporting forms protection de la vie privee, quelques may have been removed from this formulaires secondaires ont ete enleves de thesis. cette these. While these forms may be included Bien que ces formulaires aient inclus dans in the document page count, their la pagination, il n'y aura aucun contenu removal does not represent any loss manquant. of content from the thesis. 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?..