This book deals with systems of polynomial autonomous ordinary differential equations in two real variables. The emphasis is mainly qualitative, although attention is also given to more algebraic aspects as a thorough study of the center/focus problem and recent results on integrability. In the last two chapters the performant software tool P4 is introduced. From the start, differential systems are represented by vector fields enabling, in full strength, a dynamical systems approach. All essential notions, including invariant manifolds, normal forms, desingularization of singularities, index theory and limit cycles, are introduced and the main results are proved for smooth systems with the necessary specifications for analytic and polynomial systems. The book is well-suited for a first course in dynamical systems. Not only does it provide simple and appropriate proofs, but it also contains numerous exercises, and presents a survey of interesting results with accompanying references to the literature.

Authors are among the leaders in their field and well-known expositors Includes instruction manual and examples for the free software P4 (for numerical experiments on differential equations) available online Very different treatment, at somewhat higher level, from numerous other textbooks on the subject

Autorentext

Peter De Maesschalck, born in 1975, has been at Hasselt University, Belgium, for much of his career. His research focuses on slow-fast systems in low dimensional systems both from a qualitative point of view and from the point of view of asymptotic expansions. Part of his research is inspired by theoretical questions such as Hilbert's 16th problem on limit cycles of polynomial systems, another part is motivated by applications of slow-fast systems in, e.g., neurological models. Freddy Dumortier, born in 1947, emeritus professor at Hasselt University, is former president of the Belgian Mathematical Society and is currently permanent secretary of the Royal Flemish Academy of Belgium for Science and the Arts. He is the author of many papers and his main results deal with singularities and their unfolding, bifurcation theory, Liénard equations, Hilbert's 16th problem, slow-fast systems and the wave speed in reaction-diffusion equations. Robert Roussarie,born in 1944, is emeritus professor of the University of Bourgogne-Franche Comté. After a career at the CNRS he was professor at the Institut de Mathématique de Bourgogne. He worked on the theory of foliations, of singularities in differential geometry, bifurcations of vector fields and finally slow-fast systems. He also contributed to applied research on ferro-resonance in electrical networks, systems of ecological populations, systems in control theory and free interface problems in combustion theory.

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