Geometric Configurations of Singularities of Planar Polynomial Differential Systems : A Global Classification in the Quadratic Case

This book addresses the global study of finite and infinite singularities of planar polynomial differential systems, with special emphasis on quadratic systems. While results covering the degenerate cases of singularities of quadratic systems have been published elsewhere, the proofs for the remaining harder cases were lengthier. This book covers all cases, with half of the content focusing on the last non-degenerate ones. The book contains the complete bifurcation diagram, in the 12-parameter space, of global geometrical configurations of singularities of quadratic systems. The authors’ results provide - for the first time - global information on all singularities of quadratic systems in invariant form and their bifurcations. In addition, a link to a very helpful software package is included. With the help of this software, the study of the algebraic bifurcations becomes much more efficient and less time-consuming. Given its scope, the book will appeal to specialists on polynomial differential systems, pure and applied mathematicians who need to study bifurcation diagrams of families of such systems, Ph.D. students, and postdoctoral fellows. Contents 6 Preface 10 Acknowledgements 11 Part I Polynomial differential systems with emphasis on the quadratic ones 12 Chapter 1 Introduction 13 1.1 Preliminaries 13 1.2 Problems on planar polynomial differential systems 16 1.2.1 The problem of the center 16 1.2.2 Research arising from the work of Darboux and the problem on algebraic integrability 16 1.2.3 The second part of Hilbert's 16th problem 17 1.2.4 The general niteness problem for limit cycles or the existential Hilbert's 16th problem 19 1.2.5 The infinitesimal Hilbert's 16th problem and the Hilbert–Arnold problem 20 1.3 The contents of this book 20 Chapter 2 Survey of results on quadratic differential systems 27 2.1 Brief history of quadratic differential systems 27 2.2 Some basic results obtained for quadratic differential systems 30 2.3 Study of some subclasses of the family of quadratic differential systems 33 2.4 Algebraic limit cycles in quadratic systems 36 2.5 Finiteness problems for quadratic differential systems 37 2.5.1 Basic concepts and results needed for studying the general finiteness problem 37 2.5.2 The general finiteness problem for quadratic differential systems 38 2.5.3 Application of Roussarie's ideas for the quadratic case 39 2.5.4 The infinitesimal Hilbert's 16th problem and the Hilbert–Arnold problem 42 2.5.5 The infinitesimal Hilbert's 16th problem for quadratic differential systems 43 2.6 The initial steps in the global theory of quadratic diferential systems 44 Chapter 3 Singularities of polynomial differential systems 47 3.1 Compactification on the Poincaré sphere, Poincaré disc and projective plane 47 3.2 Classical definitions 51 3.3 New definitions 53 3.4 The blow-up technique 55 3.4.1 The polar blow-up 55 3.4.2 The blow-up using rational functions 56 3.4.3 The blow-up technique using only one direction 60 3.5 The borsec concept 74 3.6 Equivalence relations for singularities and for geometrical configurations of singularities of planar polynomial vector fields 85 3.7 Notations for singularities of polynomial differential systems 91 3.7.1 Elemental singularities 91 3.7.2 Non-elemental singularities 92 3.7.2.1 Semi-elemental singularities 92 3.7.2.2 Nilpotent singularities 93 3.7.2.3 Intricate singularities 93 3.7.3 Lack of singularities and complex singularities 95 3.7.4 In nite number of singularities 96 3.7.4.1 Line at infinity filled up with singularities 96 3.7.4.2 Degenerate systems 96 3.7.4.3 Degenerate systems with non-isolated singular points at infinity, isolated on the line at infinity 97 3.7.4.4 Degenerate systems with the line at infinity filled up with singularities 99 Chapter 4 Invariants in mathematical classification problems 101 4.1 Basic concepts 101 4.2 Classification problems on planar polynomial vector fields 104 4.2.1 Equivalence relations for polynomial vector fields 104 4.2.2 Classifications of some families of polynomial vector fields 106 Chapter 5 Invariant theory of planar polynomial vector fields 109 5.1 Classical invariant theory 109 5.2 The work of Sibirschi's school and collaborators in invariant theory of planar polynomial vector fields 114 5.3 Basic concepts 119 5.3.1 Group actions on polynomial vector fields 120 5.3.2 Definition of invariant polynomials for polynomial differential systems 120 5.3.3 Assembling multiplicities of singularities in divisors of the line at infinity and in zero-cycles of the plane 123 5.3.4 Construction and geometric meaning of several basic invariant polynomials 124 5.4 Invariant polynomials associated to geometrical configurations of singularities for quadratic differential systems 126 5.4.1 Building blocks for the construction of the invariant polynomials needed for the classification theorems of QS 127 5.4.2 The set of all invariant polynomials which classify geometrically the global configurations of singularities in QS 130 5.4.3 The influence of complex singularities in the study of the geometrical global configurations of singularities in QS 141 Chapter 6 Main results on classifications of finite, infinite and weak singularities in quadratic systems 143 6.1 Finite singularities 143 6.2 Finite weak singularities 154 6.3 Singularities of QS with an integrable saddle 155 6.4 Infinite singularities 158 Chapter 7 Classifications of quadratic systems with special singularities 173 7.1 A finite star node 174 7.1.1 Conditions for the existence of at least one finite star node 175 7.1.2 Configurations of singularities with a finite star node 178 7.2 An integrable saddle 189 7.3 A center 207 7.4 A star node at infinity and another special singularity 219 7.5 Three finite special singularities 231 7.6 A weak focus of order two or three 245 7.6.1 A weak focus of order two 246 7.6.2 A weak focus of order three 255 7.7 A weak saddle of order three 260 Part II Classifications of quadratic systems with special singularities 271 Chapter 8 Quadratic systems with finite singularities of total multiplicity at most one 272 8.1 Systems without nite singularities 272 8.2 Systems with exactly one singularity 276 Chapter 9 Quadratic systems with finite singularities of total multiplicity two 290 9.1 Exactly one finite singularity 290 9.2 Two distinct real singularities 299 9.3 Two distinct complex singularities 332 Chapter 10 Quadratic systems with finite singularities of total multiplicity three 337 10.1 Exactly one singularity 337 10.2 Exactly two distinct singularities 342 10.3 Exactly three distinct singularities 353 10.3.1 Systems with zero-cycle DC2(S) = p + q + r 354 10.3.2 Systems with zero-cycle DC2(S) = p + qc + rc 379 Chapter 11 Quadratic systems with finite singularities of total multiplicity four 396 11.1 Exactly one singularity 397 11.2 Exactly two distinct singularities 405 11.2.1 One triple and one simple real singularities 405 11.2.2 Two double real singularities 435 11.2.3 Two double complex singularities 440 11.3 Exactly three distinct singularities 444 11.3.1 One double and two elemental real singularities 444 11.3.2 One double real and two elemental complex singularities 491 11.4 Exactly four distinct nite singularities 496 11.4.1 Four real elemental singularities 496 11.4.2 Two real and two complex elemental singularities 553 11.4.3 Four complex elemental nite singularities 615 Chapter 12 Degenerate quadratic systems (mf = ∞) 619 Chapter 13 Conclusions 634 13.1 New concepts 634 13.2 The classical versus the new way 635 13.3 An algorithm to study the singularities of quadratic differential systems 636 13.4 The topological con gurations of singularities 637 13.5 The study of the quadratic differential systems modulo limit cycles 637 Appendix A Table of notation 640 Appendix B Manual of Mathematica tools 649 B.1 How to initiate the program 650 B.2 Notation 650 B.3 Examples 652 Bibliography 672 Index 700 This book addresses the global study of finite and infinite singularities of planar polynomial differential systems, with special emphasis on quadratic systems. While results covering the degenerate cases of singularities of quadratic systems have been published elsewhere, the proofs for the remaining harder cases were lengthier. This book covers all cases, with half of the content focusing on the last non-degenerate ones. The book contains the complete bifurcation diagram, in the 12-parameter space, of global geometrical configurations of singularities of quadratic systems. The authors' results provide - for the first time - global information on all singularities of quadratic systems in invariant form and their bifurcations. In addition, a link to a very helpful software package is included. With the help of this software, the study of the algebraic bifurcations becomes much more efficient and less time-consuming. Given its scope, the book will appeal to specialists on polynomial differential systems, pure and applied mathematicians who need to study bifurcation diagrams of families of such systems, Ph. D. students, and postdoctoral fellows