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Lyapunov Exponents and Invariant Measures of Dynamic Systems

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Abstract

According to the multiplicative ergodic theorem, Lyapunov exponents are numerically evaluated from time series of simulated dynamic systems. Introducing polar coordinates, one can split off the stationary parts of the system solution which determine the Lyapunov exponents by mean values performed in the time domain. This inifinte time integration can be reduced to a finite integral by means of associated invariant measures which are defined on the periodic ranges of the system angles.

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References

  1. Oseldec, V.I., A multiplicative ergodic theorem, Lyapunov characteristic numbers for dynamical systems, Trans. Moscow Math. Soc. 19 (1968) 197–231.

    Google Scholar 

  2. Khasminskii, R.Z., Necessary and sufficient conditions for asymptotic stability of linear stochastic systems. Theory Prob. Appl., 12 (1967) 144–147.

    Google Scholar 

  3. Coddington, E.A., Levinson, N., Theory of ordinary differential equations. McGraw-Hill, New York, 1955.

    MATH  Google Scholar 

  4. Wedig, W., Vom Chaos zur Ordnung, Mitteilungen GAMM (R. Mennicken ed.), ISSN 0936–7195, 2 (1989) 3–31.

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  5. Arnold L., Stochastic Differential Equations. Wiley, New York, 1974.

    MATH  Google Scholar 

  6. Wedig, W., Invariant measures and Lyapunov exponents of equilibria and limit cycles. Oberwolfach-Tagung 1990 on Lyapunov Exponents (L.Arnold, J.P. Eckmann).

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  7. Wedig, W., Dynamic stability of beams under axial forces–Lyapunov exponents for general fluctuating loads. Eurodyn ‘80, Conference on Structural Dynamics, Preprint of Proceedings, Ruhr-University Bochum, SFB 151, Vol. 1 (1990) 57–64.

    MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Institute for Technical Mechanics, University of Karlsruhe, D-7500, Karlsruhe 1, BRD

    Prof. Dr.-Ing. W. V. Wedig

Authors
  1. Prof. Dr.-Ing. W. V. Wedig
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Editor information

Editors and Affiliations

  1. Abteilung für Numerik, Universität Ulm, Oberer Eselsberg, D-7900, Ulm, Germany

    R. Seydel

  2. Institut für Physikalische Chemie, Universität Würzburg, Marcusstr. 9/11, D-8700, Würzburg, Germany

    F. W. Schneider

  3. Mathematisches Institut, Universität Köln, Weyertal 88, D-5000, Köln, Germany

    T. Küpper

  4. Institut für Mechanik, Technische Universität Wien, Wiedner Hauptstr. 8-10, A-1040, Wien, Austria

    H. Troger

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© 1991 Birkhäuser Verlag Basel

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Wedig, W.V. (1991). Lyapunov Exponents and Invariant Measures of Dynamic Systems. In: Seydel, R., Schneider, F.W., Küpper, T., Troger, H. (eds) Bifurcation and Chaos: Analysis, Algorithms, Applications. International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série Internationale d’Analyse Numérique, vol 97. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-7004-7_48

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  • DOI: https://doi.org/10.1007/978-3-0348-7004-7_48

  • Publisher Name: Birkhäuser Basel

  • Print ISBN: 978-3-0348-7006-1

  • Online ISBN: 978-3-0348-7004-7

  • eBook Packages: Springer Book Archive

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