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Qualitative Theory of Planar Differential Systems

Freddy Dumortier, Jaume Llibre, and Joan C. Artés
Springer Verlag
Publication Date: 
Number of Pages: 
[Reviewed by
William J. Satzer
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Qualitative Theory of Planar Differential Systems is a graduate-level introduction to systems of polynomial autonomous differential equations in two real variables. The emphasis throughout is on the vector field associated with a differential system and its phase portrait in the plane.

Many of the concepts and results presented are applicable in higher dimensional Euclidean space and on manifolds, but the authors restrict themselves completely to two dimensions. Clearly there is advantage in developing the ideas in the setting where it is easiest to visualize the results. They also avoid discussions of global analytic solutions, preferring to focus on the more practical issue of identifying and understanding qualitative behavior. Bifurcation of solutions is mentioned only briefly.

The authors say in their introduction that numerical analysis of the differential system combined with graphical representation is an essential element necessary for understanding the phase portrait. This does not mean that numerical analysis is discussed anywhere in the book; there are many pictures of phase portraits throughout, but there is no explicit numerical work anywhere.

The basic results for differential systems are presented in Chapter 1. These include existence and uniqueness of solutions as well as continuity with respect to initial conditions. Next come definitions of α- and ω-limit sets and a proof of the Poincare-Bendixson theorem that characterizes them in the plane. This chapter also introduces Lyapunov functions (for analysis of stability), describes analysis of local behavior near periodic orbits and singular points, and develops the idea of the separatrix.

Much of the rest of the book (roughly four and a half chapters) is devoted to the study of singular points of two-dimensional differential systems. The basic tool used for this is a change of variable called a blow-up. Later on, the authors present basic results on the theory of limit cycles, briefly discuss bifurcations of limit cycles, and then quickly introduce the important notion of structural stability.

The last two chapters describe a computer program developed by the authors (and available as freeware) that implements in software several of the tools developed throughout the text. The program is designed to produce a phase portrait for any planar polynomial differential system.

This text treats the basic results of the qualitative theory with competence and clarity. Much of the material on singular points is perhaps of more interest to a specialist than a neophyte, but the material of the text is well-integrated and readily accessible to graduate students or especially capable advanced undergraduates.

Bill Satzer ( is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.

 Basic Results on the Qualitative Theory of Differential Equations.- Normal Forms and Elementary Singularities.- Desingularization of Non-elementary Singularities. Centers and Lyapunov Constants. - Poincaré and Poincaré-Lyapunov Compactification.- Indices of Planar Singular Points.- Limit Cycles and Structural Stability.- Integrability and Algebaric Solutions in Polynomial Vector Fields.- Polynomial Planar Phase Portraits.- Examples for Running P4. - Bibliography.