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Birkhauser Geometrische Methoden in der Theorie der gewöhnlichen Differentialgleichungen

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lOsung von Singularitiiten, Liesche Gruppen, Newton-Diagramme) einerseits und den naturwissenschaftlichen Anwendungen andererseits. Die Theorie der partiellen Differentialgleichungen erster Ordnung wird mit Hilfe der natiirlichen Kontaktstruktur in der Mannigfaltigkeit der l-Jets von Funktionen untersucht. Nebenbei werden dabei die notwendigen Elemente der Geometrie der Kontaktstruktur dargelegt, die die ganze Theorie unabhiingig von anderen Quellen machen. Einen entscheidenden Teil des Buches nehmen die gewohnlich qualitativ genann ten Methoden ein. Die jiingste Entwicklung der von H. POINCARE begriindeten quali tativen Theorie der Differentialgleichungen flihrte zum Verstiindnis dessen, daB genauso, wie einzelne Differentialgleichungen im allgemeinen nicht vollstiindig inte grierbar sind, auch die qualitative Untersuchung gewisser allgemeiner Differential gleichungen mit mehrdimensionalem Phasenraum unmoglich ist. In diesem Zusam menhang wird die Analyse einer Differentialgleichung vom Standpunkt der Struktur stabilitiit; d. h. der Stabilitiit des qualitativen Bildes in Hinsicht auf kleine . Anderun gen der Differentialgleichung, im Kapitel 3 behandelt. Es werden die Hauptresul tate, die nach den ersten Arbeiten von A. A. ANDRONOV und L. S. PONTRJAGIN er zielt wurden, dargestellt: die Grundlagen der Theorie der strukturstabilen Anosov Systeme, deren Trajektorien alle exponentiell instabil sind, und der Satz von SMALE iiber die Nichtdichtheit der Menge der strukturstabilen Systeme. Es wird dann weiter die Bedeutung dieser mathematischen Entdeckungen flir die Anwendungen diskutiert (die Rede ist dabei von der Beschreibung stabiler chaotischer Bewe glmgsregimes, beispielweise Turbulenzen). Zu den stiirksten und am meisten verwendeten Methoden der Untersuchung von Differentialgleichungen gehoren verschiedene asymptotische Methoden.

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Format
paperback
Domain
Amazon UK
Publication Date
28 December 2011
Listed Since
21 December 2012

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