##
**Qualitative theory of planar differential systems.**
*(English)*
Zbl 1110.34002

Universitext. Berlin: Springer (ISBN 3-540-32893-9/pbk). xvi, 298 p. (2006).

This textbook, written by well-known scientists in the field of the qualitative theory of ordinary differential equations, presents a comprehensive introduction to fundamental and essential topics of real planar differential autonomous systems. From the beginning, differential systems are represented by using the vector field approach.

The book is divided into ten chapters. In Chapter 1, the fundamental notions and the basic results on the qualitative theory of differential equations with special emphasis on planar case are introduced. Chapter 2 deals with the study of the elementary singular points and also provides complete information about invariant manifolds as well as normal forms for such singularities. Chapter 3 contains the basic tool for studying nonelementary singularities of a differential system in the plane. Chapter 4 provides one of the best of algorithms currently available for distinguishing between a focus and a center. In Chapter 5, the authors introduce the so-called Poincaré-Lyapunov sphere, which is based on a construction of more abstract nature than Poincaré sphere. Chapter 6 treats singular points by using Poincaré and Hopf results on their indices.

Chapter 7 discusses the most basic results on limit cycles concerning their existence, uniqueness, configuration, multiplicity, stability and bifurcations.

Chapter 8 is devoted to the existence of first integrals for complex planar polynomial vector fields by means of the Darbouxian theory of integrability. Some recent results are included. In the last two chapters, the authors present the computer software tool \(P_4\) based on the tools introduced in the previous chapters and provide several examples about its use. Each chapter starts with a brief summary of its content. Almost all chapters end with a series of appropriate exercises and some bibliographic comments. All main results are proved for smooth systems with the necessary specifications for analytic and polynomial cases. The emphasis is mainly qualitative, although attention is also given to more algebraic aspects. There is an extensive list of references. The monograph is well written and contains a lot of illustrations and examples. It will be useful for students, teachers and researchers.

The book is divided into ten chapters. In Chapter 1, the fundamental notions and the basic results on the qualitative theory of differential equations with special emphasis on planar case are introduced. Chapter 2 deals with the study of the elementary singular points and also provides complete information about invariant manifolds as well as normal forms for such singularities. Chapter 3 contains the basic tool for studying nonelementary singularities of a differential system in the plane. Chapter 4 provides one of the best of algorithms currently available for distinguishing between a focus and a center. In Chapter 5, the authors introduce the so-called Poincaré-Lyapunov sphere, which is based on a construction of more abstract nature than Poincaré sphere. Chapter 6 treats singular points by using Poincaré and Hopf results on their indices.

Chapter 7 discusses the most basic results on limit cycles concerning their existence, uniqueness, configuration, multiplicity, stability and bifurcations.

Chapter 8 is devoted to the existence of first integrals for complex planar polynomial vector fields by means of the Darbouxian theory of integrability. Some recent results are included. In the last two chapters, the authors present the computer software tool \(P_4\) based on the tools introduced in the previous chapters and provide several examples about its use. Each chapter starts with a brief summary of its content. Almost all chapters end with a series of appropriate exercises and some bibliographic comments. All main results are proved for smooth systems with the necessary specifications for analytic and polynomial cases. The emphasis is mainly qualitative, although attention is also given to more algebraic aspects. There is an extensive list of references. The monograph is well written and contains a lot of illustrations and examples. It will be useful for students, teachers and researchers.

Reviewer: Alexander Grin (Grodno)