Skip to main content
SpringerLink
Log in

Introduction to Chaos: Phenomenon, Structure of Spectra and Diffusion

  • Chapter
Multicritical Phenomena

Part of the book series: NATO ASI Series ((NSSB,volume 106))

  • 144 Accesses

Abstract

In these lectures we present a brief introduction into a dynamical quality which is often called deterministic chaos. It is observed in many physical and other systems in which three or more order parameters or macroscopic variables xα(t) competitively determine the time behavior. The equations of motion

$${\dot x_\alpha } = {f_\alpha }(x,\alpha ),\alpha = 1,2,...,d$$
((1))

are coupled ordinary differential equations (o.d.e.s), depending on an externally controlled parameter a. If there is only one variable x1(t) the stationary solution is time independent (or goes to infinity). If there are two variables x1(t), x2(t) limit cycle behavior is possible besides the time independent steady state (or xα to infinity). The periodic (although in general not sinusoidal) oscillation is selfgenerated and not externally imposed.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. E.N. Lorenz, J.Atm.Sci. 20, 130 (1963)

    Article  ADS  Google Scholar 

  2. B.A. Huberman, J.P.Crutchfield, Phys. Rev. Lett. 43, 1743 (1979)

    Article  ADS  Google Scholar 

  3. Lord Rayleigh, Phil. Mag. 15,229 (1883); 16, 50 (1883); 24, 145 (1887)

    Google Scholar 

  4. S. Grossmann, S. Thomae, Z. Naturforsch. 32a, 1353 (1977)

    MathSciNet  ADS  Google Scholar 

  5. M.J. Feigenbaum, J. Stat. Phys. 19, 25 (1978)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  6. P. Coullet, C. Tresser, J. Physique (Paris), Colloque 39, C5-25(1978)

    Google Scholar 

  7. T. Geisel, J. Nierwetberg, Phys. Rev. Lett. 48, 7 (1982)

    Article  MathSciNet  ADS  Google Scholar 

  8. S. Grossmann, H. Fujisaka, Phys. Rev. A26, 1779 (1982)

    ADS  Google Scholar 

  9. M. Schell, S. Fraser, R. Kapral, Phys. Rev. A26, 504 (1982)

    MathSciNet  ADS  Google Scholar 

  10. B.V. Chirikov, Phys. Rep. 52, 263 (1979)

    Article  MathSciNet  ADS  Google Scholar 

  11. G. Ahlers, Phys. Rev. Lett. 33, 1185 (1974)

    Article  ADS  Google Scholar 

  12. A. Libchaber, C. Laroche, S. Fauve, J. Physique (Paris), Lettre 43, L-211 (1982)

    Article  Google Scholar 

  13. R. Helleman, in: Fundamental Problems in Statistical Mechanics, Ed. E.G.D. Cohen, North Holland,Amsterdam etc. 1980, p165 P. Collet, J.-P. Eckmann, H. Koch, Physica 3D, 457 (1981) J.M. Greene, R.S. Mc Kay, F. Vivaldi, M.J. Feigenbaum, Physica 3D,468 (1981)

    Google Scholar 

  14. A. Libchaber, J. Maurer, J. Physique (Paris), Colloque, C3-51(1980)

    Google Scholar 

  15. J.P. Gollub, S.V. Benson, J. Fluid Mech. 100, 449 (1980)

    Article  ADS  Google Scholar 

  16. M. Giglio, S. Musatti, U. Perini, Phys. Rev. Lett. 47, 243 (1981)

    Article  ADS  Google Scholar 

  17. J.P. Gollub, H. Swinney, Phys. Rev. Lett. 35, 927 (1975)

    Article  ADS  Google Scholar 

  18. R.P. Fenstermacher, H.L. Swinney, J.P. Gollub, J. Fluid Mech. 94, 103 (1979)

    Article  ADS  Google Scholar 

  19. Yu. N. Belyaev, A.A. Monakhov, S.A. Shcherbakov, I.M. Yavorskaya, JETP-Lett. 29, 295 (1979)

    ADS  Google Scholar 

  20. O.E. Rössler, Z. Naturforsch. 31a, 1168 (1976); 1664 (1976) O.E. Rbssler, K. Wegmann, Nature 271, 89 (1978)

    Article  Google Scholar 

  21. L.F. Olsen, H. Degn, Nature 267, 177 (1977)

    Article  ADS  Google Scholar 

  22. R.A. Schmitz, K.R. Graziani, J.L. Hudson, J. Chem. Phys. 67, 3040 (1977)

    Google Scholar 

  23. J.C. Roux, A. Rossi, S. Bachelard, C. Vidal, Phys. Letters 77A, 391 (1980); J.S. Turner, J.C. Roux, W.D. McCormick, H.L. Swinney, Phys. Letters 85A, 9 (1981)

    Google Scholar 

  24. P.S. Linsay, Phys. Rev. Lett. 47, 1349 (1981)

    Article  ADS  Google Scholar 

  25. J. Testa, J. Perez, C. Jeffries, Phys. Rev. Lett. 48, 714 (1982)

    Article  MathSciNet  ADS  Google Scholar 

  26. W. Lauterborn, W. Meyer-I1se, Diplomarbeit, Göttingen, 1982

    Google Scholar 

  27. W. Lauterborn, E. Cramer, Phys. Rev. Lett. 47, 1445 (1981)

    Article  ADS  Google Scholar 

  28. H. Haken, Phys. Letters 53A, 77 (1975); 62A, 133 (1977)

    Google Scholar 

  29. H.M. Gibbs, F.A. Hopf, D.L. Kaplan, R.L. Shoemaker, Phys. Rev. Lett. 46, 474 (1981)

    Article  ADS  Google Scholar 

  30. F.T. Arecchi, R. Meucci, G. Puccioni, J. Tredicce, Phys. Rev. Lett. 49, 1217 (1982)

    Article  ADS  Google Scholar 

  31. R. Keolian, I. Rudnick, L.A. Turkevich, S.J. Putterman, J.A. Rudnick, Phys. Rev. Lett. 47, 1133 (1981)

    Article  ADS  Google Scholar 

  32. C.W. Smith, M.J. Tejwani, D.A. Farris, Phys. Rev. Lett. 48, 492 (1982)

    Google Scholar 

  33. D. Meier, R. Holzner, B. Derighetti, E. Brun, in:Evolution of Chaos and Order, Ed. H. Haken, Synergetics Symposium 1982, Springer: Berlin etc. 1982

    Google Scholar 

  34. B.A. Huberman, J.P. Crutchfield, N.H. Packard, Appl. Phys. Lett. 37, 750 (1980)

    Article  ADS  Google Scholar 

  35. R.Y. Chiao, P.T. Parrish, J. Appl. Phys. 47, 2639 (1976)

    Article  ADS  Google Scholar 

  36. C. Herring, B.A. Huberman, Appl. Phys. Lett. 36, 975 (1980)

    Article  ADS  Google Scholar 

  37. D.A. Russell, E. Ott, Phys. Fluids 24, 1976 (1981)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  38. J.D. Farmer, E. Ott, J.A. Yorke, Physica D, 1982/83

    Google Scholar 

  39. J. Kaplan, J. Yorke, in: Funct. Dif f Equations etc., Eds. H.O. Peitgen, H.O. Walter, Springer: Berlin etc., 1978, p. 228

    Google Scholar 

  40. B. Mandelbrot, Fractals: Form, Chance, and Dimension, Freeman: San Francisco, 1977

    Google Scholar 

  41. P. Grassberger, I. Procaccia, Measuring the Strangeness of Attractors, Preprint 1982

    Google Scholar 

  42. J. Heldstab, H. Thomas.,, T. Geisel, G. Radons, Z.Phys.B,1983 to appear

    Google Scholar 

  43. N.H. Packard, J.P. Crutchfield, J.D. Farmer, R.S. Shaw, Phys. Rev. Lett. 45,712 (1980); H. Froehling, J.P. Crutchfield, D. Farmer, N.H. Packard, R. Shaw, Physica D, 605 (1981)

    Google Scholar 

  44. D. Singer, Siam J. Appl. Math. 35, 260 (1978)

    MATH  Google Scholar 

  45. M. Misiurewicz, Absolutely continuous measures for certain maps of an interval, Publ. Math. IHES, 1980

    Google Scholar 

  46. G. Mayer-Kress, H. Haken, Physica D, 1983, to appear

    Google Scholar 

  47. G. Mayer-Kress, H. Haken, in: Evolution of Order and Chaos, Int. Symp. on Synergetics, Elmau, Ed. H. Haken, Springer: Berlin, etc. 1982

    Google Scholar 

  48. J.P. Crutchfield, B.A. Huberman, Phys. Lett. 77A, 407 (1980)

    MathSciNet  ADS  Google Scholar 

  49. G. Mayer-Kress, H. Haken, J. Stat. Phys. 26, 149 (1982)

    Article  MathSciNet  ADS  Google Scholar 

  50. J.P. Crutchfield, J.D. Farmer, B.A. Huberman, Phys. Rep. 92, 45 (1982)

    MathSciNet  Google Scholar 

  51. A. Lasota, J.A. Yorke, Trans. Am. Math Soc. 186, 481 (1973)

    Article  MathSciNet  Google Scholar 

  52. T.Y. Li, J.A. Yorke, Trans. Am. Math Soc. 235, 183 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  53. T.Y. Li, J.A. Yorke, Amer. Math Monthly 82, 985 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  54. A.N. Sarkovskii, Ukr. Mat. Z. 16, 61 (1964)

    MathSciNet  Google Scholar 

  55. J.J. Kozak, M.K. Musho, M.D. Hatlee, Phys. Rev. Lett. 49, 1801 (1982)

    Article  MathSciNet  ADS  Google Scholar 

  56. S.J. Shenker, L.P. Kadanoff, J. Phys.A:Math.Gen.14, L 23 (1981)

    Google Scholar 

  57. B.A. Huberman, A.B. Zisook, Phys. Rev. Lett. 46, 626 (1981)

    Article  MathSciNet  ADS  Google Scholar 

  58. S. Thomae, S. Grossmann, Phys. Lett. 83A, 181 (1981)

    ADS  Google Scholar 

  59. M.J. Feigenbaum, Phys. Letters 74A, 375 (1979)

    MathSciNet  ADS  Google Scholar 

  60. M.J. Feigenbaum, in: Nonlinear Phenomena in Chemical Dynamics, Eds. C. Vidal, A. Pacault, Proc. Int. Conf. Bordeaux, Sept. 7-11, 1981, Springer: Berlin etc., 1981, p. 95

    Google Scholar 

  61. S. Thomae, S. Grossmann, J. Stat Phys. 26, 485 (1981)

    Article  MathSciNet  ADS  Google Scholar 

  62. P. Grassberger, J. Stat. Phys. 26, 173 (1981)

    Google Scholar 

  63. H. Mori, B.-C. So, T. Ose, Progr Theor Phys. 66, 1266 (1981)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  64. H. Mori, in:Nonlinear Phenomena in Chemical Dynamics, Eds. C. Vidal A. Pacault, Proc. Int. Conf. Bordeaux, Sept. 7-11, 1981, Springer: Berlin etc., 1981, p. 88

    Google Scholar 

  65. H. Shigematsu, H. Mori, T. Yoshida, H. Okamoto, two preprints, Kyushu Univ., 1982

    Google Scholar 

  66. H. Fujisaka, S. Grossmann, Z. Phys. B48, 261 (1982)

    Article  MathSciNet  ADS  Google Scholar 

  67. Y. Pomeau, P.Manneville, Phys. Lett. 75A, 1(1979); Comm. Math Phys. 74, 189 (1980)

    Article  MathSciNet  Google Scholar 

  68. G. Mayer-Kress, H. Haken, Phys. Lett. 82A, 151 (1981)

    MathSciNet  ADS  Google Scholar 

  69. J.-P. Eckmann, L. Thomas, P. Wittwer, J. Phys-A: Math. Gen. 14, 3153 (1981)

    Google Scholar 

  70. J.E. Hirsch, B.A. Huberman, D.J. Scalapino, Phys. Rev. A25, 519 (1982)

    ADS  Google Scholar 

  71. A.S. Monin, A.M. Yaglom: Statistical fluid mechanics, Vol. I, II, MIT Press: Cambridge, Mass., London, 1971

    Google Scholar 

  72. H. Fujisaka e tal., to be published

    Google Scholar 

  73. L.F. Richardson: Weather prediction by numerical process,, Cambridge University Press 1922 (reprinted 1965 )

    Google Scholar 

  74. S. Grossmann e tal., to be published Mathematical aspects are treated in:

    Google Scholar 

  75. P. Collet, J. -P. Eckmann, Iterated Maps on the Interval as Dynamical Systems, Birkhauser, Basel, 1980

    Google Scholar 

  76. I. Gumowski, C. Mira, Recurrences and Discrete Dynamic Systems, Springer, Berlin, 1980

    MATH  Google Scholar 

  77. G. Targonski, Topics in Iteration Theory, Vandenhock and Ruprecht, Götingen, 1981

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Fachbereich Physik, Philipps-Universität, Renthof 6, D-3550, Marburg, Germany

    Siegfried Grossmann

  2. Institut für Festkörperforschung der KFA, D-5170, Jülich, Germany

    Stefan Thomae

Authors
  1. Siegfried Grossmann
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Stefan Thomae
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institut Laue-Langevin, Grenoble, France

    R. Pynn

  2. Institute for Energy Technology, Kjeller, Norway

    A. Skjeltorp

Rights and permissions

Reprints and permissions

Copyright information

© 1991 kalayaan-hanger Publishing

About this chapter

Cite this chapter

Grossmann, S., Thomae, S. (1991). Introduction to Chaos: Phenomenon, Structure of Spectra and Diffusion. In: Pynn, R., Skjeltorp, A. (eds) Multicritical Phenomena. NATO ASI Series, vol 106. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-2741-7_41

Download citation

  • DOI: https://doi.org/10.1007/978-1-4613-2741-7_41

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4612-9698-0

  • Online ISBN: 978-1-4613-2741-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation