PHASE PORTRAITS OF PLANAR QUADRATIC SYSTEMS

PHASE PORTRAITS OF PLANAR QUADRATIC SYSTEMS

Mathematics and Its Applications Managing Editor: M. HAZEWINKEL Centre for Mathematics and Computer Science, Amsterdam, The Netherlands Volume 583

PHASE PORTRAITS OF PLANAR QUADRATIC SYSTEMS By JOHN REYN Delft University of Technology, The Netherlands Springer

Library of Congress Control Number: 2006927053 ISBN-10: 0-387-30413-4 e-isbn: 0-387-35215-5 ISBN-13: 978-0-387-30413-7 e-isbn: 978-0-387-35215-2 Printed on acid-free paper. AMS Subject Classifications: 34C05, 34C07, 34C15 2007 Springer Science+Business Media, LLC All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed in the United States of America. 987654321 springer.com

To my wife: Marie Therese

Contents PREFACE 1 INTRODUCTION 1 2 CRITICAL POINTS IN QUADRATIC SYSTEMS 9 2.1 Multiplicity of critical points 9 2.2 Poincare index of critical points 17 2.3 Types of critical points in quadratic systems 19 2.3.1 Finite critical points 19 2.3.2 Infinite critical points 26 3 ISOCLINES, CRITICAL POINTS AND CLASSES OF QUADRATIC SYSTEMS 31 3.1 Introduction 31 3.2 Pencil of isoclines, central conic and degenerate isoclines... 33 3.3 Central conic and finite multiplicity of a quadratic system.. 35 3.4 Quadratic systems with m/= 4 37 3.4.1 Systems with four finite critical points 37 3.4.2 Systems with three finite critical points 48 3.4.3 Systems with two finite critical points 51 3.4.4 Systems with one finite critical point 54 3.4.5 The 65 classes in quadratic systems with finite multiplicity 4 58 3.5 Quadratic systems with m/= 3 63 3.5.1 Central conic 63 3.5.2 Systems with three finite critical points 64 3.5.3 Systems with two finite critical points 67 3.5.4 Systems with one finite critical point 69

Vlll 3.5.5 The 36 classes in quadratic systems with finite multiplicity 3 69 3.6 Quadratic systems with rrif 2 72 3.6.1 Systems with one transversally non-hyperbolic infinite critical point 72 3.6.2 Systems with two transversally non-hyperbolic infinite critical points 75 3.6.3 The 48 classes in quadratic systems with finite multiplicity 2 82 3.7 Quadratic systems with rrif=l 83 3.7.1 Systems with one transversally non-hyperbolic infinite critical point 83 3.7.2 Systems with two transversally non-hyperbolic infinite critical points 85 3.7.3 Concluding remark 88 3.7.4 The 19 classes in quadratic systems with finite multiplicity 1 88 3.8 Quadratic systems with rrif= 0 89 3.8.1 Systems with one transversally non-hyperbolic infinite critical point 89 3.8.2 Systems with two transversally non-hyperbolic infinite critical points 90 3.8.3 The 5 classes in quadratic systems with finite multiplicity 0 95 4 ANALYZING PHASE PORTRAITS OF QUADRATIC SYSTEMS 97 4.1 Introduction 97 4.2 Flow over straight lines 99 4.3 General properties of quadratic systems 104 4.3.1 Critical points of center or focus type 104 4.3.2 Closed orbits 105 5 PHASE PORTRAITS OF QUADRATIC SYSTEMS IN THE CLASS ra/=0 109 5.1 Introduction 109 5.2 One transversally non-hyperbolic critical point 110 5.3 Two transversally non-hyperbolic critical points 112

IX 5.3.1 Two real distinct transversally non-hyperbolic infinite critical points 112 5.3.2 Two coinciding transversally non-hyperbolic infinite critical points 115 5.3.3 Two complex transversally non-hyperbolic infinite critical points 116 5.4 Systems with infinitely many infinite critical points 116 5.5 Conclusion 117 6 QUADRATIC SYSTEMS WITH CENTER POINTS 121 6.1 Integrability and center conditions 121 6.2 Phase portraits for quadratic systems with center points...124 6.2.1 The symmetric case: 124 6.2.2 The case with three orbital lines: 131 6.2.3 The Hamiltonian case: 135 6.2.4 The lonely case: 138 6.2.5 Theorem 6.1 139 6.3 Invariant theory for quadratic systems with center points... 140 6.4 Limit cycles and separatrix cycles 142 6.4.1 Separatrix cycles in quadratic systems with center points 142 6.4.2 Limit cycles and center points 145 7 LIMIT CYCLES IN QUADRATIC SYSTEMS 147 7.1 Introduction 147 7.2 General remarks on limit cycles, quadratic systems 148 7.3 Limit cycles in particular systems 156 7.3.1 Quadratic systems with a weak critical point 156 7.3.2 Quadratic systems with algebraic solutions 166 7.4 Limit cycle distribution over two nests 164 7.5 Bifurcation of limit cycles 174 7.5.1 Bifurcation of limit cycles from critical points 175 7.5.2 Bifurcation of limit cycles from quadratic systems with center points 176 7.5.3 Bifurcation of limit cycles from other convex closed curves 181

X 8 PHASE PORTRAITS OF QUADRATIC SYSTEMS IN THE CLASS m f =l 187 8.1 Introduction 187 8.2 One transversally non-hyperbolic critical point 188 8.2.1 Systems with a fourth order infinite critical point...188 8.2.2 Systems with a fifth order infinite critical point... 192 8.2.3 Systems with a sixth order infinite critical point... 193 8.2.4 Systems with infinitely many infinite critical points.. 194 8.3 Two transversally non-hyperbolic critical points 195 8.3.1 Two real, different transversally non-hyperbolic infinite critical points 195 8.3.2 Two coinciding transversally non-hyperbolic infinite critical points 201 8.4 Conclusions 204 9 PHASE PORTRAITS OF QUADRATIC SYSTEMS IN THE CLASS ra/=2 205 9.1 Introduction 205 9.2 One transversally non-hyperbolic critical point 206 9.2.1 Systems with a third order infinite critical point... 206 9.2.2 Systems with a fourth order infinite critical point... 231 9.2.3 Systems with a fifth order infinite critical point... 233 9.2.4 Systems with infinitely many infinite critical points.. 234 9.2.5 Conclusion 235 9.3 Two transversally non-hyperbolic critical points 235 9.3.1 Two real, different transversally non-hyperbolic infinite critical points 235 9.3.2 Two coinciding transversally non-hyperbolic infinite critical points 249 9.3.3 Two complex transversally non-hyperbolic infinite critical points 253 9.3.4 Conclusion 257 9.4 Conclusions 257 10 PHASE PORTRAITS OF QUADRATIC SYSTEMS IN THE CLASS ra/=3 259 10.1 Introduction 259

XI 10.2 Phase portraits of systems with a degenerate infinite critical point 260 10.2.1 Phase portraits of systems with a fourth order infinite critical point 260 10.2.2 Phase portraits of systems with a third order infinite critical point 261 10.2.3 Conclusion 263 10.3 Phase portraits of systems with a saddle node at infinity... 263 10.3.1 Systems with finite index i/ = 1 265 10.3.2 Systems with finite index if = 1 271 10.3.3 Conclusion 273 10.4 Phase portraits with infinite critical points at infinity 273 11 PHASE PORTRAITS OF QUADRATIC SYSTEMS IN THE CLASS m/=4 277 11.1 Introduction 277 11.2 Systems with finite index if = 2 278 11.2.1 Systems with less than four real finite critical points.. 278 11.2.2 The system e~ 1 e~ 1 e~ 1 e 1 E 1 E 1 E 1 having four real finite critical points 281 11.3 Systems with finite index if 0 284 11.3.1 Systems with a fourth order finite critical point....285 11.3.2 Systems with a third order finite critical point 287 11.3.3 Systems with a finite critical point of the second order mlot c 2 291 11.3.4 Systems with four finite critical points 296 11.4 Systems with finite index if = 2 299 11.4.1 Systems with less than four finite critical points... 299 11.4.2 Systems with four finite critical points 301 11.5 Structurally stable quadratic systems 303 BIBLIOGRAPHY 305 INDEX 333

xiii Preface Solutions of ordinary differential equations usually can not be expressed in terms of well known functions if these equations are nonlinear. As a result numerical and asymptotic methods are developed to obtain approximations to solutions. An alternative approach is furnished by the qualitative theory of differential equations, which seeks to find properties of solutions without actually solving these equations. As such the geometric topological theory of plane autonomous systems x = P(x,y),y Q(x,y) due to Poincare and Bendixson is well known and by now has assumed an almost definite form. A first step within these systems away from the linear systems leads to the case where P and Q are relative prime polynomials of degree at most 2 which are not both linear, and thus to the system x = a 00 + a m x + aoiy + a 2 ox 2 + a n xy + a 02 y 2, (0.0.1) y = b 00 + b 10 x + b 01 y + b 20 x 2 + b n xy + b 02 y 2 (0.0.2) for the functions x = x(t), y = y(t), where a^,bij (i,j = 0,1,2) are real (or sometimes complex) constants. In a survey paper on these systems in 1966 by W.A.Coppel these systems were called quadratic systems, by which they have been referred to ever since. Apart from a variety of problems in various fields of applications that lead to these systems they are also of theoretical interest, being a relatively simple example to study complicated non linear phenomena such as limit cycle behaviour. It then helps to consider the traces of the solutions x = x(t),y = y(t) in the x,y plane, being the orbits that constitute the phase portrait in the x,y (phase) plane. Although some examples of phase portraits of quadratic systems can already be found in the work of Poincare, it seems that the first paper dealing exclusively with these systems was published in 1904 by Biichel, although it was mainly a collection of examples. In the remainder of the twentieth century an increasing flow of results produced over a thousand papers; the accumulated number of papers that appeared over those years is illustrated in the accompanying figure. The present book tries to give a presentation of the advance of our knowledge of phase portraits of quadratic systems as a result of this flow of papers, thereby paying attention to the historical development of the subject. This

xiv is also expressed by the chronological structure of the list of references at the end of the book. Constructing this presentation raises the question of how to order what is known about these portraits. In this book a particular ordering in classes, using the notions of finite and infinite multiplicity and finite and infinite index, is presented and classifications of phase portraits for various classes are given, using the well known methods of phase plane analysis. 1000 H 900 H 800 - Rest of the World 700 -\ 600 -\ 500 -\ Soviet Union 400-300 - 200-100 -\ 25 "I I I ^T I I I I I I r 1900 04 10 20 30 40 50 60 70 80 90 2000 Accumulated number of papers on quadratic systems published in the twentieth century In Chapter 1, a review is given of the basic notions of phase portraits, keeping a focus on quadratic systems. It is primarily meant as an

XV introduction for readers with insufficient knowledge of the Poincare Bendixson theory. In Chapter 2, critical points in quadratic systems are discussed, being the first basic elements needed to construct a phase portrait. The notion of finite multiplicity rrif is introduced,being the sum of the multiplicities of the finite (real or complex) critical points, and it is shown that in quadratic systems 0 < rrif < 4, if the number of critical points is finite. Finite multiplicity is chosen as the first ordering principle to define classes of quadratic systems. The partition of the coefficient space corresponding to the value of rrif is given.infinite critical points and infinite multiplicity rrii are defined, where rrii is the sum of the multiplicities of the infinite critical points; if the number of critical points is finite, then rrif + rrii = 7. Further ordering principles are the finite index if ( 2 < if < 2) and the infinite index U ( 1 < U < 3) based on the Poincare index definition for a critical point, and it is shown that if + i% = 1. The possible types of finite and infinite critical (real or complex) points are discussed. In Chapter 3, the identification of critical points with base points of the pencil of conies, being isoclines, leads to an extensive study of these isoclines, wherein degenerate isoclines and central conies obtain special attention. A classification of possible degenerate isocline combinations and character of the central conic leads, within each class characterized by a given value of rrif (0 < rrif < 4) and if ( 2 < if < 2), to possible combinations of finite and infinite critical points that define a further subdivision in classes. Following this procedure, 173 classes are obtained to represent all quadratic systems. In Chapter 4, general properties of quadratic systems and results obtained from studying particular classes,characterized by a specific property, are indicated in order to be used in classification of the 173 classes. The possible types of flow in a quadratic system over a straight line are classified and the results applied to derive general properties of quadratic systems. In Chapter 5, the phase portraits in class rrif = 0 are classified; there exist 30 possible phase portraits in rrif = 0.

xvi In Chapter 6, quadratic systems with a center point are considered and imbedded in the 173 classes. Chapter 7 gives an extensive historic sketch of what has become known about limit cycles in quadratic systems, starting at an elementary level of knowledge about the limit cycle notion. In Chapter 8, the phase portraits in the class rrif = 1 are classified; there exist 38 possible phase portraits in class rrif = 1. In Chapter 9, the phase portraits in the class rrif = 2 are classified; there exist 230 possible phase portraits in class rrif = 2. In Chapter 10, the phase portraits in the class rrif = 3 are considered. In classes with a degenerate infinite critical point there exist 141 possible phase portraits. For classes with a semi- elementary infinite critical point, 269 possible separatrix structures are given without solving the limit cycle problem. In Chapter 11, the phase portraits in class rrif = 4 are discussed; the classification in rrif = 4 is far from being complete. Many people have contributed to the development of my interest in quadratic systems and, directly or indirectly, to the motivation to write this book. They are too numerous to mention by name. I hope, in return, that they derive much pleasure in reading it. I would like to express my thanks to those persons working in Delft University of Technology, Faculty of Technical Mathematics, who helped me to produce the book on the computer. I would like to thank Mr.Wu for installing the Latex program on my home computer and Mr. Kees Lemmens for doing the same, now also introducing the Linux operating system and helping me to start working with the computer to construct an electronic version of the book. My special thanks goes to Mr. Ruud Sommerhalder, who was tireless helping me to overcome the problems that kept coming up during this work. A lot of work still remains to be done to complete the collection of all phase portraits of quadratic systems. If this book helps in promoting the achievement of this goal, it has served its purpose. John W. Reyn