Advertisement

Synergetics pp 333-349 | Cite as

Chaos

  • Hermann Haken
Part of the Springer Series in Synergetics book series (SSSYN, volume 1)

Abstract

Sometimes scientists like to use dramatic words of ordinary language in their science and to attribute to them a technical meaning. We already saw an example in Thorn’s theory of “catastrophes”. In this chapter we become acquainted with the term “chaos”. The word in its technical sense refers to irregular motion. In previous chapters we encountered numerous examples for regular motions, for instance an entirely periodic oscillation, or the regular occurrence of spikes with well-defined time intervals. On the other hand, in the chapters about Brownian motion and random processes we treated examples where an irregular motion occurs due to random, i. e., in principle unpredictable, causes.

Keywords

Rayleigh Number Chaotic Motion Strange Attractor Period Doubling Representative Point 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

What is Chaos

  1. For mathematically rigorous treatments of examples of chaos by means of mappings and other topological methods see S. Smale: Bull. A. M. S. 73, 747 (1967)zbMATHCrossRefMathSciNetGoogle Scholar
  2. For mathematically rigorous treatments of examples of chaos by means of mappings and other topological methods see T. Y. Li, J. A. Yorke: Am. Math. Monthly 82, 985 (1975)zbMATHCrossRefMathSciNetGoogle Scholar
  3. For mathematically rigorous treatments of examples of chaos by means of mappings and other topological methods see D. Ruelle, F. Takens: Commun. math. Phys. 20, 167 (1971)ADSzbMATHCrossRefMathSciNetGoogle Scholar

The Lorenz Model. Motivation and Realization

  1. E. N. Lorenz: J. Atmospheric Sci. 20, 130 (1963)ADSCrossRefGoogle Scholar
  2. E. N. Lorenz: J. Atmospheric Sci. 20, 448 (1963)ADSCrossRefGoogle Scholar
  3. Historically, the first papers showing a “strange attractor”. For further treatments of this model see J. B. McLaughlin, P. C. Martin: Phys. Rev. A12, 186 (1975)ADSGoogle Scholar
  4. Historically, the first papers showing a “strange attractor”. For further treatments of this model see M. Lücke: J. Stat. Phys. 15, 455 (1976)ADSCrossRefGoogle Scholar
  5. Historically, the first papers showing a “strange attractor”. For further treatments of this model see C. T. Sparrow: The Lorenz Equations: Bifurcations, Chaos and Strange Attractors (Springer, Berlin-Heidelberg-New York 1982)zbMATHCrossRefGoogle Scholar
  6. For the laser fluid analogy presented in this chapter see H. Haken: Phys. Lett. 53 A, 77 (1975)ADSGoogle Scholar

How Chaos Occurs

  1. H. Haken, A. Wunderlin: Phys. Lett. 62 A, 133 (1977)ADSMathSciNetGoogle Scholar

Chaos and the Failure of the Slaving Principle

  1. H. Haken, J. Zorell: UnpublishedGoogle Scholar

Correlation Function and Frequency Distribution

  1. M. Lücke: J. Stat. Phys. 15, 455 (1976)ADSCrossRefGoogle Scholar
  2. Y. Aizawa, I. Shimada: Preprint 1977Google Scholar

Further Examples of Chaotic Motion

  1. Three Body Problem: H. Poincaré: Les méthodes nouvelles de la méchanique céleste. Gauthier-Villars, Paris (1892/99), Reprint (Dover Publ., New York 1960)Google Scholar
  2. For electronic devices especially Khaikin’s “universal circuit” see A. A. Andronov, A. A. Vitt, S. E. Khaikin: Theory of Oscillators (Pergamon Press, Oxford-London-Edinburgh-New York-Toronto-Paris-Frankfurt 1966)zbMATHGoogle Scholar
  3. Gunn Oscillator: K. Nakamura: Progr. Theoret. Phys. 57, 1874 (1977)ADSCrossRefGoogle Scholar
  4. Numerous chemical reaction models (without diffusion) have been treated byGoogle Scholar
  5. O. E. Roessler. For a summary and list of reference consultGoogle Scholar
  6. O. E. Roessler: In Synergetics, A Workshop, ed. by H. Haken (Springer, Berlin-Heidelberg-New York, 1977)Google Scholar
  7. For chemical reaction models including diffusion see Y. Kuramoto, T. Yamada: Progr. Theoret. Phys. 56, 679 (1976)ADSCrossRefMathSciNetGoogle Scholar
  8. For chemical reaction models including diffusion see T. Yamada, Y. Kuramoto: Progr. Theoret. Phys. 56, 681 (1976)ADSCrossRefMathSciNetGoogle Scholar
  9. Modulated chemical reactions have been trated by K. Tomita, T. Kai, F. Hikami: Progr. Theoret. Phys. 57, 1159 (1977)ADSCrossRefGoogle Scholar
  10. For experimental evidence of chaos in chemical reactions see R. A. Schmitz, K. R. Graziani, J. L. Hudson: J. Chem. Phys. 67, 3040 (1977);ADSCrossRefGoogle Scholar
  11. O. E. Roessler, to be publishedGoogle Scholar
  12. Earth magnetic field: J. A. Jacobs: Phys. Reports 26, 183 (1976) with further referencesADSCrossRefGoogle Scholar
  13. Population dynamics: R. M. May: Nature 261, 459 (1976)ADSCrossRefGoogle Scholar
  14. Review articles: M. I. Rabinovich: Sov. Phys. Usp. 21, 443 (1978)ADSCrossRefGoogle Scholar
  15. A. S. Monin: Sov. Phys. Usp. 21, 429 (1978)ADSCrossRefGoogle Scholar
  16. D. Ruelle: La Recherche N° 108, Février (1980)Google Scholar
  17. Some fundamental works on period doubling are S. Grossmann, S. Thomae: Z. Naturforsch. A32, 1353 (1977)ADSMathSciNetGoogle Scholar
  18. This paper deals with the logistic map given in the text. The universal behavior of period doubling was discovered by M. J. Feigenbaum: J. Stat. Phys. 19, 25 (1978):ADSzbMATHCrossRefMathSciNetGoogle Scholar
  19. This paper deals with the logistic map given in the text. The universal behavior of period doubling was discovered by M. J. Feigenbaum: Phys. Lett. A74, 375 (1979)ADSMathSciNetGoogle Scholar
  20. An extensive presentation of later results, as well as many further references, are given in P. Collet, J. P. Eckmann: Iterated Maps on the Interval as Dynamical System (Birkhäuser, Boston 1980)Google Scholar
  21. Conference proceedings dealing with chaos: L. Garrido (ed.): Dynamical Systems and Chaos, Lecture Notes Phys., Vol. 179 (Springer, Berlin, Heidelberg, New York 1983)Google Scholar
  22. H. Haken (ed.): Chaos and Order in Nature, Springer Ser. Synergetics, Vol. 11 (Springer, Berlin Heidelberg, New York 1981)Google Scholar
  23. H. Haken (ed.): Evolution of Order and Chaos, Springer Ser. Synergetics, Vol. 17 (Springer, Berlin, Heidelberg, New York 1982)Google Scholar
  24. The influence of fluctuations on period doubling has been studied by the following authors: G. Mayer-Kress, H. Haken: J. Stat. Phys. 24, 345 (1981)CrossRefMathSciNetGoogle Scholar
  25. The influence of fluctuations on period doubling has been studied by the following authors: J. P. Crutchfield, B. A. Huberman: Phys. Lett. A77, 407 (1980)ADSMathSciNetGoogle Scholar
  26. The influence of fluctuations on period doubling has been studied by the following authors: A. Zippelius, M. Lücke: J. Stat. Phys. 24, 345 (1981)ADSCrossRefGoogle Scholar
  27. The influence of fluctuations on period doubling has been studied by the following authors: J. P. Crutchfield, M. Nauenberg, J. Rudnick: Phys. Rev. Lett. 46, 933 (1981)ADSCrossRefMathSciNetGoogle Scholar
  28. The influence of fluctuations on period doubling has been studied by the following authors: B. Shraiman, C. E. Wayne, P. C. Martin: Phys. Rev. Lett. 64, 935 (1981)ADSCrossRefMathSciNetGoogle Scholar
  29. The corresponding Kolmogorov equation is established and discussed in H. Haken, G. Mayer-Kress: Z. Phys. B43. 185 (1981)ADSCrossRefMathSciNetGoogle Scholar
  30. The corresponding Kolmogorov equation is established and discussed in H. Haken, A. Wunderlin: Z. Phys. B46, 181 (1982)ADSCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1983

Authors and Affiliations

  • Hermann Haken
    • 1
  1. 1.Institut für Theoretische PhysikUniversität StuttgartStuttgart 80Fed. Rep. of Germany

Personalised recommendations