Skip to main content
SpringerLink
Log in

Strange Attractors

  • Chapter
Foundations of Synergetics II

Part of the book series: Springer Series in Synergetics ((SSSYN,volume 52))

Abstract

In this chapter we begin the analysis of dissipative dynamical systems. In contrast to the Hamiltonian systems discussed earlier, these systems compress the phase volume. In dissipative dynamical systems (almost) all trajectories are asymtotically attracted with time to some limit sets of zero volume which are called attractors. Simple attractors are points, lines and (hyper)surfaces in the phase space. However, much more complex strange attractor are also possible.

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. A.J. Lichtenberg, M.A. Lieberman: Regular and Stochastic Motion (Springer, Berlin, Heidelberg 1983 )

    Google Scholar 

  2. O.E. Lanford: “Strange attractors and turbulence”, in Hydrodynamic Instabilities and Transition to Turbulence, ed. by H.L. Swinney, J.P. Gollub ( Springer, Berlin, Heidelberg 1981 ) pp. 7–31

    Google Scholar 

  3. J.P. Eckmann: Rev. Mod. Phys. 53, 643–654 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  4. J. Guckenheimer, P. Holmes: Nonlinear Oscillations, Dynamical Systems and Bifurcation of Vector Fields (Springer, Berlin, Heidelberg 1983 )

    Google Scholar 

  5. V.I. Arnold, V.S. Afraimovich, Yu.S. Ilyashenko, L.P. Shilnikov: “Bifurcation theory”, in Dynamical Systems, Vol. 5, ed. by V. I. Arnold (Springer, Berlin, Heidelberg 1989 )

    Google Scholar 

  6. V.S. Afraimovich: “Attractors”, in Nonlinear Waves 1. Dynamics and Evolution, ed. by J. Engelbrecht, A.V. Gaponov-Grekhov, M.I. Rabinovich ( Springer, Berlin, Heidelberg 1989 ) pp. 14–28

    Google Scholar 

  7. J.M.T. Thompson, H.B. Stewart: Nonlinear Dynamics and Chaos ( Wiley, New York 1986 )

    MATH  Google Scholar 

  8. Ya.G. Sinai: “Stochasticity of dynamical Systems”, in Nonlinear Waves, ed. by A.V. Gaponov-Grekhov, M.I. Rabinovich ( Nauka, Moscow 1979 ) pp. 192–212

    Google Scholar 

  9. J. Milnor: Commun. Math. Phys. 99, 177–196 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  10. S. Newhouse: Ann. N.Y. Acad. Sci. 357, 292–299 (1980)

    Article  Google Scholar 

  11. D. Ruelle, F. Takens: Commun. Math. Phys. 20, 167–192 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  12. R. Shaw: Z. Naturforsch. A36, 80–112 (1981)

    MATH  MathSciNet  Google Scholar 

  13. V.S. Afraimovich, L.P. Shilnikov: “On strange attractors and quasiattractors”, in Nonlinear Dynamics and Turbulence ( Pitman, Boston 1983 ) pp. 1–34

    Google Scholar 

  14. E.N. Lorenz: J. Atmos. Sci. 20, 130–141 (1963)

    Article  Google Scholar 

  15. C. Sparrow: The Lorenz Equations: Bifurcations, Chaos and Strange Attractors (Springer, Berlin, Heidelberg 1982 )

    Google Scholar 

  16. L.A. Bunimovich, Ya.G. Sinai: “Stochasticity of the attractor in the Lorenz model”, in Nonlinear Waves, ed. by A.V. Gaponov-Grekhov, M.I. Rabinovich ( Nauka, Moscow 1979 ) pp. 212–226

    Google Scholar 

  17. V.S. Afraimovich, V.V. Bykov, L.P. Shilnikov: Soy. Phys.-Dokl. 22, 563 (1977)

    Google Scholar 

  18. J.A. Yorke, E.D. Yorke: J. Stat. Phys. 21, 263–277 (1979)

    Article  MathSciNet  Google Scholar 

  19. O.E. Lanford: “Computer pictures of the Lorenz attractor”, in Lecture Notes in Mathematics, Vol. 615 ( Springer, Berlin, Heidelberg 1977 ) pp. 113–116

    Google Scholar 

  20. S. Smale: Bull. Amer. Math. Soc. 73, 747–817 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  21. R.F. Williams: Global Analysis. Amer. Math. Soc. 14, 361–393 (1970)

    Google Scholar 

  22. A.N. Kolmogorov: Dokl. Akad. Nauk SSSR 119, 861–864 (1958)

    MATH  MathSciNet  Google Scholar 

  23. A.N. Kolmogorov: Dokl. Akad Nauk SSSR 124, 754–755 (1959)

    MATH  MathSciNet  Google Scholar 

  24. Ya.G. Sinai: Dokl. Akad. Nauk SSSR 124, 768–771 (1959)

    MATH  MathSciNet  Google Scholar 

  25. Yu.L. Klimontovich: Statistical Physics ( Harwood, New York 1986 )

    Google Scholar 

  26. G. Benettin, L. Galgani, J.M. Strelcyn: Phys. Rev. A 14, 2338–2345 (1976)

    Article  Google Scholar 

  27. G. Benettin, C. Froeschle, J.P. Scheidecker: Phys. Rev. A 19, 2454–2460 (1979)

    Article  MathSciNet  Google Scholar 

  28. P. Hartman: Ordinary Differential Equations ( Wiley, New York 1964 )

    MATH  Google Scholar 

  29. I. Shimada, T. Nagascima: Progr. Theor. Phys. 61, 1605–1616 (1979)

    Article  MATH  Google Scholar 

  30. S.D. Feit: Commun. Math. Phys. 61, 249–260 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  31. G. Benettin, L. Galgani, A. Giogilli, J.M. Strelcyn: Meccanica 15, 9–31 (1980)

    Article  MATH  Google Scholar 

  32. J.S. Turner, J.-C. Roux, W.D. McCormick, H.L. Swinney: Phys. Lett. A85, 9–12 (1981)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Max-Planck-Gesellschaft, Fritz-Haber-Institut, Faradayweg 4-6, D-14195, Berlin, Germany

    Professor Alexander S. Mikhailov

  2. Department of Physics, Lomonosov Moscow State University, 117234, Moscow, Russia

    Professor Alexander Yu. Loskutov

Authors
  1. Professor Alexander S. Mikhailov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Professor Alexander Yu. Loskutov
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 1996 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Mikhailov, A.S., Loskutov, A.Y. (1996). Strange Attractors. In: Foundations of Synergetics II. Springer Series in Synergetics, vol 52. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-80196-9_3

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-80196-9_3

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-80198-3

  • Online ISBN: 978-3-642-80196-9

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation