Abstract
This paper proposes a generally applicable approach for the bifurcation analysis of nonlinear dissipative systems, described by smooth, ordinary and autonomous differential equations. The analysis is done in two stages. The first stage is the brute force calculation of Lyapunov exponents. The second stage is the more sophisticated partially analytical investigation of selected points of interest. There-fore we use a parametrized Taylor series approximation of the Poincaré map, where the analytical derivations can be performed by using computer algebra. The approximation enables the determination of the type of bifurcation, the iterative calculation of bifurcation points and stability analysis at the critical value by application of center manifold theory. The proposed approach is demonstrated by application to the Duffing oscillator in the version of Ueda.
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References
Carr, J.: ‘Applications of Centre Manifold Theory’. New York/…: Springer-Verlag, 1981.
Hsu, C. S.: ‘A Theory of Cell-to-Cell Mapping Dynamical Systems’. J. of Appl. Mech. 47, 1980, 931–939.
Kleczka, M.: ‘Zur Berechnung der Ljapunov-Exponenten und deren Bedeutung’. Stuttgart: Universität Stuttgart, Institut B für Mechanik, Studienarbeit STUD-16, 1985.
Kleczka, W.: ‘Einsatz von Computer-Algebra zur Analyse nichtlinearer dynamischer Probleme’. Stuttgart: Universität Stuttgart, Institut B für Mechanik, Studienarbeit STUD-50, 1989.
Kreuzer, E.: ‘Numerische Untersuchung nichtlinearer dynamischer Systeme’. Berlin/…: Springer-Verlag, 1987.
MACSYMA-Group of SYMBOLICS, Inc.: ‘MACSYMA Reference Manual, Version 11’. 11 Cambridge Center, Cambridge, MA 02142 (1986).
Rand, R. H.; Armbruster, D.: ‘Perturbation Methods, Bifurcation Theory and Computer Algebra’. New York/…: Springer-Verlag, 1987.
Troger, H.; Lindtner, E.; Steindl, A.: ‘Generic One-parameter Bifurcations in the Motion of a Simple Robot’. Wien: Technische Universität Wien, 1988.
Ueda, Y.: ‘Steady motions exhibited by Duffing’s equation: A picture book of regular and stochastic motions’. In: ‘New Approaches to Nonlinear Problems in Dynamics’ (Editor: P. J. Holmes ), SIAM, Philadelphia, 1980, 311–322.
Wilmers, C.: ‘Verzweigungsphänomene in mechanischen Oszillatoren’. Stuttgart: Universität Stuttgart, Institut B für Mechanik, Diplomarbeit DIPL-24, 1988.
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© 1990 Kluwer Academic Publishers
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Kleczka, M., Kleczka, W., Kreuzer, E. (1990). Bifurcation Analysis: A Combined Numerical and Analytical Approach. In: Roose, D., Dier, B.D., Spence, A. (eds) Continuation and Bifurcations: Numerical Techniques and Applications. NATO ASI Series, vol 313. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-0659-4_8
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DOI: https://doi.org/10.1007/978-94-009-0659-4_8
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-6781-2
Online ISBN: 978-94-009-0659-4
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