Qualitative Theory of Planar Differential Systems

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

FREDDY DUMORTIER is full professor at Hasselt University (Belgium), and a member of the Royal Flemish Academy of Belgium for Science and the Arts. He was a long-term visitor at different important universities and research institutes. He is the author of many papers and his main results deal with singularities and their unfolding, singular perturbations, Lienard equations and Hilbert's 16th problem.

JAUME LLIBRE is full professor at the Autonomous University of Barcelona (Spain), he is a member of the Royal Academy of Sciences and Arts of Barcelona. He was a long term visitor at different important universities and research institutes. He is the author of many papers and had a large number of Ph. D. students. His main results deal with periodic orbits, topological entropy, polynomial vector fields, Hamiltonian systems and celestial mechanics.

JOAN C. ARTES is professor at the Autonomous University of Barcelona (Spain). His main results deal with polynomial vector fields, more concretely quadratic ones. He programmed, some 20 years ago, the first version of P4 (only for quadratic systems) from which the program P4 was developed with the help of Chris Herssens and Peter De Maesschalck.

Klappentext
The book deals essentially 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: based on both algebraic manipulation and numerical calculation, this was conceived for the purpose of drawing "Polynomial Planar Phase Portraits" on part of the plane, or on a Poincaré compactification, or even on a Poincaré-Lyapunov compactification of the plane.

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 very appropriate for a first course in dynamical systems, presenting the basic notions in the study of individual two dimensional systems. Not only does it provide simple and appropriate proofs, but it also contains a lot of exercises and presents a survey of interesting results with the necessary references to the literature.

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.