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 Preface Acknowledgements Part I Polynomial differential systems with emphasis on the quadratic ones Chapter 1 Introduction 1.1 Preliminaries 1.2 Problems on planar polynomial differential systems 1.2.1 The problem of the center 1.2.2 Research arising from the work of Darboux and the problem on algebraic integrability 1.2.3 The second part of Hilbert's 16th problem 1.2.4 The general niteness problem for limit cycles or the existential Hilbert's 16th problem 1.2.5 The infinitesimal Hilbert's 16th problem and the Hilbert–Arnold problem 1.3 The contents of this book Chapter 2 Survey of results on quadratic differential systems 2.1 Brief history of quadratic differential systems 2.2 Some basic results obtained for quadratic differential systems 2.3 Study of some subclasses of the family of quadratic differential systems 2.4 Algebraic limit cycles in quadratic systems 2.5 Finiteness problems for quadratic differential systems 2.5.1 Basic concepts and results needed for studying the general finiteness problem 2.5.2 The general finiteness problem for quadratic differential systems 2.5.3 Application of Roussarie's ideas for the quadratic case 2.5.4 The infinitesimal Hilbert's 16th problem and the Hilbert–Arnold problem 2.5.5 The infinitesimal Hilbert's 16th problem for quadratic differential systems 2.6 The initial steps in the global theory of quadratic diferential systems Chapter 3 Singularities of polynomial differential systems 3.1 Compactification on the Poincaré sphere, Poincaré disc and projective plane 3.2 Classical definitions 3.3 New definitions 3.4 The blow-up technique 3.4.1 The polar blow-up 3.4.2 The blow-up using rational functions 3.4.3 The blow-up technique using only one direction 3.5 The borsec concept 3.6 Equivalence relations for singularities and for geometrical configurations of singularities of planar polynomial vector fields 3.7 Notations for singularities of polynomial differential systems 3.7.1 Elemental singularities 3.7.2 Non-elemental singularities 3.7.2.1 Semi-elemental singularities 3.7.2.2 Nilpotent singularities 3.7.2.3 Intricate singularities 3.7.3 Lack of singularities and complex singularities 3.7.4 In nite number of singularities 3.7.4.1 Line at infinity filled up with singularities 3.7.4.2 Degenerate systems 3.7.4.3 Degenerate systems with non-isolated singular points at infinity, isolated on the line at infinity 3.7.4.4 Degenerate systems with the line at infinity filled up with singularities Chapter 4 Invariants in mathematical classification problems 4.1 Basic concepts 4.2 Classification problems on planar polynomial vector fields 4.2.1 Equivalence relations for polynomial vector fields 4.2.2 Classifications of some families of polynomial vector fields Chapter 5 Invariant theory of planar polynomial vector fields 5.1 Classical invariant theory 5.2 The work of Sibirschi's school and collaborators in invariant theory of planar polynomial vector fields 5.3 Basic concepts 5.3.1 Group actions on polynomial vector fields 5.3.2 Definition of invariant polynomials for polynomial differential systems 5.3.3 Assembling multiplicities of singularities in divisors of the line at infinity and in zero-cycles of the plane 5.3.4 Construction and geometric meaning of several basic invariant polynomials 5.4 Invariant polynomials associated to geometrical configurations of singularities for quadratic differential systems 5.4.1 Building blocks for the construction of the invariant polynomials needed for the classification theorems of QS 5.4.2 The set of all invariant polynomials which classify geometrically the global configurations of singularities in QS 5.4.3 The influence of complex singularities in the study of the geometrical global configurations of singularities in QS Chapter 6 Main results on classifications of finite, infinite and weak singularities in quadratic systems 6.1 Finite singularities 6.2 Finite weak singularities 6.3 Singularities of QS with an integrable saddle 6.4 Infinite singularities Chapter 7 Classifications of quadratic systems with special singularities 7.1 A finite star node 7.1.1 Conditions for the existence of at least one finite star node 7.1.2 Configurations of singularities with a finite star node 7.2 An integrable saddle 7.3 A center 7.4 A star node at infinity and another special singularity 7.5 Three finite special singularities 7.6 A weak focus of order two or three 7.6.1 A weak focus of order two 7.6.2 A weak focus of order three 7.7 A weak saddle of order three Part II Classifications of quadratic systems with special singularities Chapter 8 Quadratic systems with finite singularities of total multiplicity at most one 8.1 Systems without nite singularities 8.2 Systems with exactly one singularity Chapter 9 Quadratic systems with finite singularities of total multiplicity two 9.1 Exactly one finite singularity 9.2 Two distinct real singularities 9.3 Two distinct complex singularities Chapter 10 Quadratic systems with finite singularities of total multiplicity three 10.1 Exactly one singularity 10.2 Exactly two distinct singularities 10.3 Exactly three distinct singularities 10.3.1 Systems with zero-cycle DC2(S) = p + q + r 10.3.2 Systems with zero-cycle DC2(S) = p + qc + rc Chapter 11 Quadratic systems with finite singularities of total multiplicity four 11.1 Exactly one singularity 11.2 Exactly two distinct singularities 11.2.1 One triple and one simple real singularities 11.2.2 Two double real singularities 11.2.3 Two double complex singularities 11.3 Exactly three distinct singularities 11.3.1 One double and two elemental real singularities 11.3.2 One double real and two elemental complex singularities 11.4 Exactly four distinct nite singularities 11.4.1 Four real elemental singularities 11.4.2 Two real and two complex elemental singularities 11.4.3 Four complex elemental nite singularities Chapter 12 Degenerate quadratic systems (mf = ∞) Chapter 13 Conclusions 13.1 New concepts 13.2 The classical versus the new way 13.3 An algorithm to study the singularities of quadratic differential systems 13.4 The topological con gurations of singularities 13.5 The study of the quadratic differential systems modulo limit cycles Appendix A Table of notation Appendix B Manual of Mathematica tools B.1 How to initiate the program B.2 Notation B.3 Examples Bibliography Index 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