Skip to main content

Feigenbaum Scenario

  • Chapter
Chaos
  • 166 Accesses

Abstract

In this chapter, we will study a very important class of dynamical systems, which is almost ideally suited as an introduction to the basic characteristics of nonlinear dynamics, namely discrete mappings. An example of such a discrete system is provided by a stroboscopic map, where a system is only observed at well-defined time steps t i . In some situations, the discrete mapping arises directly from the nature of the system, as for instance in population dynamics or kicked or billiard-type systems. In other cases, one uses a discrete approximation to the true continuous evolution.

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 69.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

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. P. Collet and J.-P. Eckmann, Iterated Maps on the Interval as Dynamical Systems (Birkhäuser, Basel 1980)

    MATH  Google Scholar 

  2. R. May, Simple mathematical models with very complicated dynamics, Nature 261 (1976) 459 (reprinted in: B.-L. Hao, Chaos (World Scientific, Singapore 1984) and P. Cvitanovic, Universality in Chaos (Adam Hilger, Bristol 1984))

    Article  ADS  Google Scholar 

  3. M. J. Feigenbaum, Universal behaviour in nonlinear systems, Los Alamos Science 1 (1980) 4 (reprinted in: B.-L. Hao, Chaos (World Scientific, Singapore 1984))

    MathSciNet  Google Scholar 

  4. H. G. Schuster, Deterministic Chaos (VCH, Weinheim 1988)

    Google Scholar 

  5. R. L. Devaney, An Introduction to Chaotic Dynamical Systems (Addison-Wesley, New York 1987)

    Google Scholar 

  6. N. Metropolis, M. L. Stein, and P. R. Stein, On finite limit sets for transformations on the unit interval, Jour. of Combinatorial Theory 15 (1973) 25 (reprinted in: B.-L. Hao, Chaos (World Scientific, Singapore 1984) and P. Cvitanovic, Universality in Chaos (Adam Hilger, Bristol 1984))

    Article  MathSciNet  MATH  Google Scholar 

  7. S. Grossmann and S. Thomae, Invariant distributions and stationary correlation functions of one-dimensional discrete processes, Z. Naturf. A 32 (1977) 1353 (reprinted in: P. Cvitanovic, Universality in Chaos (Adam Hilger, Bristol 1984))

    MathSciNet  ADS  Google Scholar 

  8. M. J. Feigenbaum, Quantitative universality for a class of nonlinear transformations, J. Stat. Phys. 19 (1978) 158 (reprinted in: B.-L. Hao, Chaos (World Scientific, Singapore 1984))

    Article  MathSciNet  Google Scholar 

  9. M. J. Feigenbaum, The universal metric properties of nonlinear transformations, J. Stat. Phys. 21 (1979) 669 (reprinted in: B.-L. Hao, Chaos (World Scientific, Singapore 1984) and P. Cvitanovic, Universality in Chaos (Adam Hilger, Bristol 1984))

    Article  MathSciNet  ADS  MATH  Google Scholar 

  10. O. E. Lanford III, A computer-assisted proof of the Feigenbaum conjectures, Bull. Am. Math. Soc. 6 (1982) 427 (reprinted in: P. Cvitanovic, Universality in Chaos (Adam Hilger, Bristol 1984))

    Article  MathSciNet  MATH  Google Scholar 

  11. A. N. Sarkovskii, Coexistence of cycles of a continuous map of a line into itself, Ukr. Mat. Z. 16 (1964) 61

    MathSciNet  Google Scholar 

  12. T.-Y. Li and J. A. Yorke, Period three implies chaos, Ann. Math. Monthly 82 (1975) 985 (reprinted in: B.-L. Hao, Chaos (World Scientific, Singapore 1984))

    Article  MathSciNet  MATH  Google Scholar 

  13. J. Frøyland, Introduction to Chaos and Coherence (IOP Publishing, Bristol 1992)

    Google Scholar 

  14. J. P. Crutchfield, J. D. Farmer, and B. A. Huberman, Fluctuations and simple chaotic dynamics, Phys. Rep. 92 (1982) 45

    Article  MathSciNet  ADS  Google Scholar 

  15. C. Bracikowski and R. Roy, Chaos in a multimode solid-state laser system, Chaos 1 (1991) 49

    Article  ADS  Google Scholar 

  16. T. Shinbrot, C. Grebogi, J. Wisdom, and J. A. Yorke, Chaos in a double pendulum, Am. J. Phys. 60 (1992) 491

    Article  ADS  Google Scholar 

  17. S. Martin, H. Leber, and W. Martienssen, Oscillatory and chaotic states of the electrical conduction in barium sodium niobate crystals, Phys. Rev. Lett. 53 (1984) 303

    Article  ADS  Google Scholar 

  18. T. M. Mello and N. M. Tufillaro, Strange attractors of a bouncing ball, Am. J. Phys. 55 (1987) 316

    Article  ADS  Google Scholar 

  19. M. T. Levinsen, The chaotic oscilloscope, Am. J. Phys. 61 (1993) 155

    Article  ADS  Google Scholar 

  20. J. C. Earnshaw and D. Haughey, Lyapunov exponents for pedestrians, Am. J. Phys. 61 (1993) 401

    Article  ADS  Google Scholar 

  21. K. Briggs, Simple experiments in chaotic dynamics, Am. J. Phys. 55 (1987) 1083

    Article  ADS  Google Scholar 

  22. H. N. Nunez Yépez, A. L. Salas Brito, C. A. Vargas, and L. A. Vincente, Chaos in a dripping faucet, Eur. J. Phys. 10 (1989) 99

    Article  Google Scholar 

  23. R. F. Cahalan, H. Leidecker, and G. D. Cahalan, Chaotic rhythms of a dripping faucet, Comput. in Phys. Jul./Aug. (1990) 368

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

Copyright information

© 1994 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Korsch, H.J., Jodl, HJ. (1994). Feigenbaum Scenario. In: Chaos. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-02991-6_9

Download citation

  • DOI: https://doi.org/10.1007/978-3-662-02991-6_9

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-02993-0

  • Online ISBN: 978-3-662-02991-6

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics