Dumortier / Llibre / Artés Qualitative Theory of Planar Differential Systems

1. Auflage 2006
ISBN: 978-3-540-32902-2
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 302 Seiten, eBook

Reihe: Universitext

Our aim is to study ordinary di?erential equations or simply di?erential s- tems in two real variables x ? = P(x,y), (0.1) y? = Q(x,y), r 2 where P and Q are C functions de?ned on an open subset U of R , with ? r=1,2,...,?,?.AsusualC standsforanalyticity.Weputspecialemphasis onto polynomial di?erential systems, i.e., on systems (0.1) where P and Q are polynomials. Instead of talking about the di?erential system (0.1), we frequently talk about its associated vector ?eld ? ? X = P(x,y) +Q(x,y) (0.2) ?x ?y 2 on U? R . This will enable a coordinate-free approach, which is typical in thetheoryofdynamicalsystems.Anotherwayexpressingthevector?eldisby writingitasX=(P,Q).Infact,wedonotdistinguishbetweenthedi?erential system (0.1) and its vector ?eld (0.2). Almost all the notions and results that we present for two-dimensional di?erential systems can be generalized to higher dimensions and manifolds; but our goal is not to present them in general, we want to develop all these notions and results in dimension 2. We would like this book to be a nice introduction to the qualitative theory of di?erential equations in the plane, providing simultaneously the major part of concepts and ideas for developing a similar theory on more general surfaces and in higher dimensions. Except in very limited cases we do not deal with bifurcations, but focus on the study of individual systems.

Zielgruppe

Graduate

Autoren/Hrsg.

Dumortier, Freddy

Llibre, Jaume

Artés, Joan C.

Inhaltsverzeichnis

Basic Results on the Qualitative Theory of Differential Equations.- Normal Forms and Elementary Singularities.- Desingularization of Nonelementary Singularities.- Centers and Lyapunov Constants.- Poincaré and Poincaré–Lyapunov 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.

Über Autor(innen)

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 16 th 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.