Skip to main content
SpringerLink
Log in
Book cover

The Theory of Chaotic Attractors pp 243–264Cite as

On the Concept of Attractor

  • Chapter

Abstract

This note proposes a definition for the concept of “attractor,” based on the probable asymptotic behavior of orbits. The definition is sufficiently broad so that every smooth compact dynamical system has at least one attractor.

This is a preview of subscription content, 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   84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   109.99
Price excludes VAT (USA)
  • Durable hardcover 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

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Milnor, J.: On the concept of attractor. Commun. Math. Phys. 99, 177–195 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  2. Smale, S.: Differentiable dynamical systems. Bull. Am. Math. Soc. 73, 747–817 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  3. Besicovitch, A.S.: A problem on topological transformation of the plane. Fundam. Math. 27, 61–65 (1937) (compare [4])

    Google Scholar 

  4. Besicovitch, A.S.: A problem on topological transformations of the plane. II. Proc. Camb. Phil. Soc. 47, 38–45 (1951)

    Article  MathSciNet  MATH  Google Scholar 

  5. Conley, C.: Isolated invariant sets and the Morse index. CBMS-NSF Reg. Conf. 38, Am. Math. Soc. 1978

    Google Scholar 

  6. Auslander, J., Bhatia, N.P., Seibert, P.: Attractors in dynamical systems. Bol. Soc. Mat. Mex. 9, 55–66 (1964)

    MathSciNet  MATH  Google Scholar 

  7. Dobrynskii, V.A., Sarkovskii, A.N.: Genericity of dynamical systems almost all orbits of which are stable under sustained perturbations. Soy. Math. Dokl. 14, 997–1005 (1973)

    Google Scholar 

  8. Hurley, M.: Attractors: persistence, and density of their basins. Trans. Am. Math. Soc. 269, 247–271 (1982) (compare [9])

    Google Scholar 

  9. Hurley, M.: Bifurcation and chain recurrence. Ergodic Theory Dyn. Syst. 3, 231–240 (1983)

    MathSciNet  MATH  Google Scholar 

  10. Newhouse, S.: Lectures on dynamical systems, pp. 1–114 of “Dynamical Systems”. Guckenheimer, Moser, Newhouse (ed.). Boston, Basel, Stuttgart: Birkhäuser 1980

    Google Scholar 

  11. Robinson, R.C., Williams, R.F.: Finite stability is not generic, pp. 451–462 of “Dynamical Systems”. Peixoto, M. (ed.). New York: Academic Press 1973

    Google Scholar 

  12. Takens, F.: Tolerance stability, pp. 293–304 of “Dynamical Systems”. Lecture Notes in Mathematics, Vol. 468, Manning, A. (ed.). Berlin, Heidelberg, New York: Springer 1975

    Google Scholar 

  13. Mayer, D., Roepstorff, G.: Strange attractors and asymptotic measures of discrete-time dissipative systems. J. Stat. Phys. 31, 309–326 (1983)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute for Advanced Study, Princeton, NJ, 08540, USA

    John Milnor

Authors
  1. John Milnor
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institute for Physical Science and Technology, University of Maryland, 20742, College Park, MD, USA

    Brian R. Hunt

  2. Department of Mathematics, Michigan State University, 48824, East Lansing, Michigan, USA

    Tien-Yien Li

  3. Department of Mathematics, University of Delaware, 19716, Newark, Delaware, USA

    Judy A. Kennedy

  4. Department of Econometrics, University of Groningen, NL-9700 AV, Groningen, The Netherlands

    Helena E. Nusse

Rights and permissions

Reprints and permissions

Copyright information

© 1985 Springer Science+Business Media New York

About this chapter

Cite this chapter

Milnor, J. (1985). On the Concept of Attractor. In: Hunt, B.R., Li, TY., Kennedy, J.A., Nusse, H.E. (eds) The Theory of Chaotic Attractors. Springer, New York, NY. https://doi.org/10.1007/978-0-387-21830-4_15

Download citation

  • DOI: https://doi.org/10.1007/978-0-387-21830-4_15

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4419-2330-1

  • Online ISBN: 978-0-387-21830-4

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation