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Titel:Homoclinic bifurcations in reversible systems
Autor: Wagenknecht, Thomas [Autor]
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Dateien vom 13.05.2004 / geändert 24.05.2005
URL für Lesezeichen:http://www.db-thueringen.de/servlets/DocumentServlet?id=1745
URN (NBN):urn:nbn:de:gbv:ilm1-2003000161
Kollektion:Dissertationen/Habilitationen
Status:Dokument veröffentlicht
Sprache:Deutsch
Dokumententyp:Dissertation
Medientyp:Text
Beitragende:Prof. Dr. Champneys, A. R. [Gutachter]
Prof. Dr. Fiedler, Bernold [Gutachter]
Prof. Dr. rer. nat. habil. Marx, Bernd [Betreuer/Doktorvater]
Stichwörter:Gewöhnliche Differentialgleichung; Homokliner Orbit; Verzweigung <Mathematik>; Reversibles System
Evaluationstyp:Für die Langzeitarchivierung vorgesehen
Dewey Decimal Classification:500 Naturwissenschaften und Mathematik » 510 Mathematik » 515 Analysis
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Beschreibungen:Zusammenfassung (dt.)

Abstract:
The thesis investigates bifurcations from homoclinic solutions of ordinary differential equations. Homoclinic solutions are characterised by approaching an equilibrium, i.e. a constant solution of a differential equation, in both positive and negative time. The thesis is devoted to the analysis of homoclinic bifurcations that originate from a change in the type of the associated equilibrium. Several scenarios are considered in the class of reversible ordinary differential equations. The main part of the thesis deals with solutions homoclinic to equilibria that themselves undergo a local bifurcation. In this process the type of the equilibrium changes from a real saddle (with real leading eigenvalues) to a saddle-centre (with a pair of imaginary eigenvalues). The interplay of local and global bifurcation effects requires a new analytical approach. By a combination of analytical and geometric techniques a description of bifurcating homoclinic solutions is derived. Thereby both purely reversible systems and systems with additional symmetry or Hamiltonian structure are considered. The second part of the thesis discusses a homoclinic bifurcation in which the associated equilibrium undergoes a transition from real saddle to complex saddle-focus (with complex leading eigenvalues). The existence of two primary homoclinic solutions forming a so-called bellows structure is assumed. Using an analytical technique known as Lin?s method results about the bifurcation of N-homoclinic orbits are derived. The theory is applied to physical problems from nonlinear optics and water wave theory as well as to two mathematical model systems. Numerical investigations confirm the general bifurcation results.
Hochschule/Fachbereich:Technische Universität Ilmenau » Fakultät für Mathematik und Naturwissenschaften
Dokument erstellt am: 13.05.2004
Dateien geändert am: 13.05.2004
Datum der Promotion: 12.12.2003