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Lyapunov exponents and invariant measures of equilibria and limit cycles

  • Chapter 5: Engineering Applications And Control Theory
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Lyapunov Exponents

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1486))

Abstract

Khasminskii's projection on circles, spheres or hyperspheres leads to the top Lyapunov exponents of dynamic systems. Provided there exists an invariant measure, the multiplicative ergodic theorem of Oseledec can be reduced to a finite integral on the projection angles. This technique is demonstrated by nonlinear deterministic systems with self-exciting terms and by linear systems with parametric excitations by white noise. The paper emphasizes different numerical methods to solve Liouville or Fokker-Planck equations and to determine the invariant measures of dynamic systems.

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References

  1. Weidenhammer, F. 1969. Biegeschwingungen des Stabes unter axial pulsierender Zufallslast. VDI-Berichte Nr. 135: 101–107.

    Google Scholar 

  2. Khasminskii, R.Z. 1967. Necessary and sufficient conditions for asymptotic stability of linear stochastic systems. Theor. Prob. and Appls. 12: 144–147.

    Article  MathSciNet  Google Scholar 

  3. Arnold, L. 1974. Stochastic Differential Equations, Theory and Applications. New York, Wiley.

    MATH  Google Scholar 

  4. Wedig, W. 1988. Simulation and analysis of mechanical systems with parameter fluctuations. To appear in Proceedings of the Oberwolfach Conference on Random Partial Differential Equations (ed. by U. Hornung), Springer, Lecture Notes in Mathematics.

    Google Scholar 

  5. Oseledec, V.I. 1968. A multiplicative ergodic theorem, Lyapunov characteristic numbers for dynamical systems. Trans. Moscow Math. Soc. 19: 197–231.

    MathSciNet  Google Scholar 

  6. Kloeden, P.E. and Platen, E. 1989. A survey of numerical methods for stochastic differential equations. Stochastic Hydrology and Hydraulics 3, 155–178.

    Article  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  8. Arnold, L. & Kliemann, W. 1981. Qualitative theory of stochastic systems. In: Probabilistic Analysis and Related Topics (ed. by A.T. Bharucha-Reid). Vol. 3, New York: Academic Press.

    Google Scholar 

  9. Xie, Wei-Chau. 1990. Lyapunov exponents and their applications in structural dynamics. Ph D thesis, supervised by S. T. Ariaratnam and presented to the University of Waterloo, Canada.

    Google Scholar 

  10. Wedig, W. 1990. Lyapunov exponents and invariant measures for isotropic limit cycles. To appear in Proceedings of the Second International Conference on Stochastic Structural Dynamics, Boca Raton, Florida, May 1990, Florida Atlantic University, Center for Applied Stochastics Research.

    Google Scholar 

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Author information

Authors and Affiliations

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

    Prof. Dr.-Ing. W. Wedig

Authors
  1. Prof. Dr.-Ing. W. Wedig
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Ludwig Arnold Hans Crauel Jean-Pierre Eckmann

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© 1991 Springer-Verlag

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Wedig, W. (1991). Lyapunov exponents and invariant measures of equilibria and limit cycles. In: Arnold, L., Crauel, H., Eckmann, JP. (eds) Lyapunov Exponents. Lecture Notes in Mathematics, vol 1486. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0086678

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  • DOI: https://doi.org/10.1007/BFb0086678

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-54662-7

  • Online ISBN: 978-3-540-46431-0

  • eBook Packages: Springer Book Archive

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