Geometric Configurations of Singularities of Planar Polynomial Differential Systems

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.

Presents novel, powerful tools for studying algebraic bifurcations in quadratic differential systems Introduces an algebra software package that will allow readers to avoid complicated calculations once they have understood the main concepts Provides methods that are highly useful for studying several large families of quadratic systems and for checking classifications made with classical tools, as well as revealing some flaws in them

Auteur
Joan C. Artés is Associate Professor at the Departament de Matemàtiques, Universitat Autònoma de Barcelona in Barcelona, Spain.Jaume Llibre is Full Professor at the Departament de Matemàtiques, Universitat Autònoma de Barcelona in Barcelona, Spain.

Dana Schlomiuk is Honorary Professor, former Full Professor at the Département de Mathématiques et de Statistiques, Université de Montréal in Montreal, Canada.

Nicolae Vulpe is Professor, Principal Researcher at the Vladimir Andrunachievici Institute of Mathematics and Computer Science in Chisinau, Moldova.

Résumé
"This excellent book is a survey of results in the qualitative study of real planar quadratic differential systems, a significant part of which was derived by the authors themselves, who are outstanding researchers in this field. ... The book ends with an extensive bibliography concerning various aspects of the topic of the presented research results. This survey will be very useful for the interested reader." (Alexander Grin, zbMATH 1493.37001, 2022)

Contenu
Part I.- Polynomial differential systems with emphasis on the quadratic ones.- 1 Introduction.- 2 Survey of results on quadratic differential systems.- 3 Singularities of polynomial differential systems.- 4 Invariants in mathematical classification problems.- 5 Invariant theory of planar polynomial vector fields.- 6 Main results on classifications of singularities in QS.- 7 Classifications of quadratic systems with special singularities.- Part II.- 8 QS with finite singularities of total multiplicity at most one.- 9 QS with finite singularities of total multiplicity two.- 10 QS with finite singularities of total multiplicity three.- 11 QS with finite singularities of total multiplicity four.- 12 Degenerate quadratic systems.- 13 Conclusions.