Abstract
Bifurcation theories for the instability of slowly evolving systems have been developed in various disciplines, and a first step is here taken towards some desirable unification. A modern account of the authors' general branching theory for discrete systems is first presented, some new features being the introduction of principal imperfections and the delineation of the important semi-symmetric points of bifurcation. This theory, embedded in a perturbation approach ideal for quantitative analysis, is complementary to the far-reaching qualitative catastrophe theory of René Thom which offers a profound topological classification of instability phenomena. For this reason, we present here a detailed correlation of the two theories.
Also presented in the paper is a survey of some fields of application ranging from classical fields such as hydrodynamics, through thermodynamics, crystallography and cosmology, to the newer domains of biology and psychology.
Zusammenfassung
Verzweigungstheorien für die Instabilität sich allmählich entwickelnder Systeme wurden in verschiedenen Disziplinen entwickelt; es wird hier der erste Schritt zu einer erwünschten Vereinheitlichung getan. Einleitend wird ein moderner Abriss der allgemeinen Verzweigungstheorie diskreter Systeme, wie sie von den Verfassern entwickelt wurde, dargestellt. Einige neue Elemente sind die Einführung von ‘Hauptimperfektionen’ und die Beschreibung halbsymmetrischer Verzweigungspunkte. Diese Theorie, verbunden mit der für eine quantitative Analyse idealen Störungsrechnung, ergänzt die weitreichende qualitative Katastrophentheorie von René Thom, die eine tiefschürfende topologische Klassifizierung der Instabilitätserscheinungen bietet. Aus diesem Grunde wird hier die Wechselbeziehung der zwei Theorien eingehend dargestellt.
Anschliessend wird eine Uebersicht einiger Anwendungsgebiete geboten—vom klassischem Feld der Hydrodynamik über die Thermodynamik, Kristallographie und Kosmologie zu den neueren Bereichen der Biologie und Psychologie.
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Thompson, J.M.T., Hunt, G.W. Towards a unified bifurcation theory. Journal of Applied Mathematics and Physics (ZAMP) 26, 581–603 (1975). https://doi.org/10.1007/BF01594031
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DOI: https://doi.org/10.1007/BF01594031