About this book
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.
- Book Title Qualitative Theory of Planar Differential Systems
- DOI https://doi.org/10.1007/978-3-540-32902-2
- Copyright Information Springer 2006
- Publisher Name Springer, Berlin, Heidelberg
- eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
- Softcover ISBN 978-3-540-32893-3
- eBook ISBN 978-3-540-32902-2
- Edition Number 1
- Number of Pages XVI, 302
- Number of Illustrations 0 b/w illustrations, 0 illustrations in colour
Ordinary Differential Equations
Dynamical Systems and Ergodic Theory
- Buy this book on publisher's site
From the reviews:
"Qualitative Theory of Planar Differential Systems is a graduate-level introduction to systems of polynomial autonomous differential equations in two real variables. … This text treats the basic results of the qualitative theory with competence and clarity. … the material of the text is well-integrated and readily accessible to graduate students or especially capable advanced undergraduates." (William J. Satzer, MathDL, December, 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. … 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." (Alexander Grin, Zentralblatt MATH, Vol. 1110 (12), 2007)
"The planar differential systems which are the subject of this book are systems of autonomous differential equations … . This book contains a wealth of information and techniques, some of it unavailable outside the research literature. … Moreover the exposition is accurate, clear, and well-motivated. … this work could serve well both as a textbook for a course in smooth dynamical systems on planar regions, and as a reference in which important tools of current research are thoroughly explained and their use illustrated." (Douglas S. Shafer, Mathematical Reviews, Issue 2007 f)