Qualitative Theory of Planar Differential Systems

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.

Sommario

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.

Info autore

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.

Riassunto

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.

Relazione

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)

Dettagli sul prodotto

• Autori: Joan C. Artés, Freddy Dumortier, Jaume Llibre
• Editore: Springer, Berlin
• Lingue: Inglese
• Formato: Tascabile
• Pubblicazione: 06.09.2006
• EAN: 9783540328933
• ISBN: 978-3-540-32893-3
• Pagine: 302
• Dimensioni: 160 mm x 17 mm x 236 mm
• Peso: 488 g
• Illustrazioni: XVI, 302 p.
• Serie: Universitext
• Categorie: Scienze naturali, medicina, informatica, tecnica > Matematica > Analisi; Analysis, B, Dynamics, Mathematics and Statistics, Ordinary Differential Equations, Dynamical Systems and Ergodic Theory, Ergodic theory, Nonlinear science, Differential equations, Dynamical systems