Skip to main content

Order and Chaos: Period-Doubling and its Chaotic Mirror

  • Chapter
Fractals for the Classroom

Abstract

Chaos theory began at the end of last century with some great initial ideas, concepts and results of the monumental French mathematician Henri Poincaré. Also the more recent path of the theory has many fascinating success stories. Probably the most beautiful and important one is the theme of this chapter. It is known as the route from order into chaos, or Feigenbaum’s universality.

… there is a God precisely because Nature itself, even in chaos, cannot proceed except in an orderly and regular manner.

Immanuel Kant

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

Access this chapter

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

eBook
USD 9.99
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
Hardcover Book
USD 54.99
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

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Reference

  1. M. J. Feigenbaum, Quantiative universality for a class of nonlinear transformations, J. Stat. Phys. 19 (1987) 25–52

    Google Scholar 

  2. P. Grassberger and I. Procaccia, Measuring the strangeness of strange attractors, Physica 9D (1983) 189–208.

    MathSciNet  MATH  Google Scholar 

  3. P. Collet and J.-P.Eckmann, Iterated Maps on the Interval as Dynamical Systems, Birkhäuse, Boston, 1980, and M. Feigenbaum, Universal behavior in nonlinear systems, Physica 7D (1983) 16–39, also in: D. Campbell and H. Rose (eds.), Order in Chaos North- Holland, Amsterdam, 1983.

    Google Scholar 

  4. S. Großman and S. Thomae, Invariant distributions and stationary correlation functions of one-dimensional discrete processes Zeitschrift Für Naturforschg. 32 (1977) 1353–1363.

    Google Scholar 

  5. R. M. May’s remarkable pater, Simple mathematical models with very complicated dynamics, Nature 261 (1976) 459–467.

    Article  Google Scholar 

  6. Y. Pomeau and P. Manneville, Intermittent transition to turbulence in dissipative dynamical systems, Commun. Math. Phys. 74 (1980) 189–197.

    Article  MathSciNet  Google Scholar 

  7. R. W. Leven, B.-P. Koch and B. Pompe, Chaos in Dissipativen Systemen, Vieweg, Braunschweig, 1989.

    MATH  Google Scholar 

  8. Grebogi, C., E. Ott, J. A. Yorke, Crises, Sudden changes in chaotic attractors, and transient chaos, Physica 7D (1983) 181–200.

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

Copyright information

© 1992 Springer-Verlag New York, Inc.

About this chapter

Cite this chapter

Peitgen, HO., Jürgens, H., Saupe, D. (1992). Order and Chaos: Period-Doubling and its Chaotic Mirror. In: Fractals for the Classroom. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-4406-6_5

Download citation

  • DOI: https://doi.org/10.1007/978-1-4612-4406-6_5

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4612-8758-2

  • Online ISBN: 978-1-4612-4406-6

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics