Qualitative Theory of Planar Differential Systems
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
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Freddy Dumortier Jaume Llibre Joan C. Artés
Qualitative Theory of Planar Differential Systems 123
Freddy Dumortier, Jaume Llibre, Joan C. Artés
Qualitative Theory of Planar Differential Systems With 123 Figures and 10 Tables
123
Freddy Dumortier Hasselt University Campus Diepenbeek Agoralaan-Gebouw D 3590 Diepenbeek, Belgium e-mail: [email protected]
Jaume©Llibre Universitat Autònoma de Barcelona Dept.©Matemátiques 08193©Cerdanyola Barcelona, Spain e-mail: [email protected]
Joan©C.©Artés Universitat Autònoma de Barcelona Dept.©Matemátiques 08193©Cerdanyola Barcelona, Spain e-mail: [email protected]
Mathematics Subject Classification (2000): 34Cxx (34C05, 34C07, 34C08, 34C14, 34C20, 34C25, 34C37, 34C41), 37Cxx (37C10, 37C15, 37C20, 37C25, 37C27, 37C29)
Library of Congress Control Number: 2006924563
ISBN-10 3-540-32893-9 Springer Berlin Heidelberg New York ISBN-13 3-540-32902-1 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com © Springer-Verlag Berlin Heidelberg 2006
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Preface
Our aim is to study ordinary differential equations or simply differential systems in two real variables x˙ = P (x, y), (0.1) y˙ = Q(x, y), where P and Q are C r functions defined on an open subset U of R2 , with r = 1, 2, . . . , ∞, ω. As usual C ω stands for analyticity. We put special emphasis onto polynomial differential systems, i.e., on systems (0.1) where P and Q are polynomials. Instead of talking about the differential system (0.1), we frequently talk about its associated vector field X = P (x, y)
∂ ∂ + Q(x, y) ∂x ∂y
(0.2)
on U ⊂ R2 . This will enable a coordinate-free approach, which is typical in the theory of dynamical systems. Another way expressing the vector field is by writing it as X = (P, Q). In fact, we do not distinguish between the differential system (0.1) and its vector field (0.2). Almost all the notions and results that we present for two-dimensional differential systems can be generalized to higher dimensions and manifolds; but our goal is
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