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Ingenieur-Archiv

, Volume 55, Issue 4, pp 285–294 | Cite as

Analysis of chaotic systems using the cell mapping approach

  • E. J. Kreuzer
Article

Summary

Chaotic motions in deterministic nonlinear systems are an important topic both from a theoretical and a practical point of view. In particular, there have been many studies of systems which yield bounded nonperiodic trajectories converging to attractors of a rather complicated nature, so-called strange attractors. Their existence was demonstrated in a class of nonlinear oscillators with periodic forcing which occur in electric circuit theory and mechanics. The determination of the domain of attraction of such attractors, depending on the parameters, is an interesting problem. It is shown, that the cell mapping approach, i.e., a discrete version of a Poincaré map, represents a very efficient method for analyzing this problem.

Keywords

Neural Network Complex System Nonlinear System Information Theory Nonlinear Dynamics 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Analyse chaotischer Systeme mit der Zellabbildungsmethode

Übersicht

Chaotische Bewegungen in deterministischen nichtlinearen Systemen sind sowohl unter theoretischen als auch praktischen Gesichtspunkten von Bedeutung. Viele Untersuchungen haben sich im besonderen mit Systemen beschäftigt, deren Bewegung durch nichtperiodische Trajektorien gekennzeichnet ist, die zu komplizierten Attraktoren, sogenannten seltsamen Attraktoren, konvergieren. Die Existenz dieser Attraktoren wurde an einer Reihe von nichtlinearen, periodisch erregten Oszillatoren demonstriert, welche in elektrischen Schwingkreisen und der Mechanik auftreten. Die Bestimmung des Einzugsgebietes solcher Attraktoren in Abhängigkeit von den Systemparametern ist ein interessantes Problem. Es wird gezeigt, daß die Zellabbildungsmethode, eine diskrete Version einer Poincaré-Abbildung, ein sehr effizientes Verfahren zur Analyse dieses Problems darstellt.

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References

  1. 1.
    Feigenbaum, M.: The transition to aperiodic behaviour in turbulent systems. Commun. Math. Phys. 77 (1980) 65–86Google Scholar
  2. 2.
    Ruelle, D.; Takens, F.: On the nature of turbulence. Commun. Math. Phys. 20 (1971) 167–192Google Scholar
  3. 3.
    Shaw, R.: Strange attractors, chaotic behavior, and information flow. Z. Naturforschung 36 (1981) 80–112Google Scholar
  4. 4.
    Eckmann, J.-P.: Roads to turbulence in dissipative systems. Reviews of Modern Phys. 53 (1981) 643 to 654Google Scholar
  5. 5.
    Ott, E.: Strange attractors and chaotic motions of dynamical systems. Review of Modern Phys. 53 (1981) 655–671Google Scholar
  6. 6.
    Cayley, A.: Application of the Newton-Fourier method to an imaginary root of an equation. Quaterly J. of Pure and Appl. Math. XVI (1879) 179–185Google Scholar
  7. 7.
    Julia, G.: Mémoire sur l'itération des fonctions rationnelles. J. de Math. pures et appliquées, sér. 8.1 (1918) 47–245Google Scholar
  8. 8.
    Hsu, C. S.: A theory of cell-to-cell mapping dynamical systems. J. of Appl. Mech. 47 (1980) 931–939Google Scholar
  9. 9.
    Hsu, C. S.: A generalized theory of cell-to-cell mapping for nonlinear dynamical systems. J. of Appl. Mech. 48 (1981) 634–642Google Scholar
  10. 10.
    Rannou, F.: Numerical study of discrete plane area-preserving mappings. Astron. and Astrophys. 31 (1974) 289–301Google Scholar
  11. 11.
    Bestle, D.; Kreuzer, E.: Analyse von Grenzzyklen mit der Zellabbildungsmethode Z. Angew. Math. Mech. 65 (1985) 4, T29-T32Google Scholar
  12. 12.
    Kreuzer, E. J.: Domains of attraction in systems with limit cycles. In: Proc. of German-Japanese Seminar on Nonlinear Problems in Dynamical Systems, Universität Stuttgart 1984, 8.1–8.24Google Scholar
  13. 13.
    Guckenheimer, T.; Holmes, P.: Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Berlin, Heidelberg, New York: Springer 1983Google Scholar
  14. 14.
    Hénon, M.: A two-dimensional mapping with a strange attractor. Commun. Math. Phys. 50 (1976) 69–77Google Scholar
  15. 15.
    Lorenz, E. N.: Deterministic nonperiodic flow. J. Atmos. Sci. 20 (1983) 130–141Google Scholar
  16. 16.
    Ruelle, D.: Strange attractors. The Mathematical Intelligencer 2 (1980) 126–137Google Scholar
  17. 17.
    Feit, S. D.: Characteristic exponents and strange attractors. Commun. Math. Phys. 61 (1978) 249–260Google Scholar
  18. 18.
    Curry, J. H.: On the Hénon transformation. Commun. Math. Phys. 68 (1979) 129–140Google Scholar
  19. 19.
    Simó, C.: On the Hénon-Pomeau attractor. J. of Stat. Phys. 21 (1979) 465–494Google Scholar
  20. 20.
    Franceshini, V.; Russo, L.: Stable and unstable manifolds of the Hénon mapping. J. of Stat. Phys. 22 (1981) 757–769Google Scholar
  21. 21.
    Oseledec, V. I.: A multiplicativ ergodic theorem: Liapunov characteristic numbers for dynamical systems. Trans. Moscow Math. Soc. 19 (1968) 197–231Google Scholar
  22. 22.
    Ruelle, D.: Sensitive dependence on initial condition and turbulent behavior of dynamical systems. In: Bifurcation Theory and Applications in Scientific Disciplines. O. Gurel and O. E. Rössler (eds.) New York: New York Acad. of Sciences 1979, 408–446Google Scholar
  23. 23.
    Hsu, C. S.; Kim, M. C.: Statistics of strange attractors by generalized cell mapping. To appear in J. of Stat. Phys. (1985)Google Scholar

Copyright information

© Springer-Verlag 1985

Authors and Affiliations

  • E. J. Kreuzer
    • 1
  1. 1.Institut B für MechanikUniversität StuttgartStuttgart 80Bundesrepublik Deutschland

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