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A Route to Chaos

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Systems Analysis and Simulation I

Part of the book series: Advances in Simulation ((ADVS.SIMULATION,volume 1))

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Abstract

The route to chaos of the coupled logistic map is studied. We observe a flip bifurcation from a stable fixed point to a stable period-2 orbit which is followed by a Hopf bifurcation, quasiperiodic behaviour and periodic orbits. At the end of the route the iteration scheme tends to a fascinating strange attractor looking like the Eiffel tower.

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References

  1. Beau, W., W. Metzler and A. Überla: The Route to Chaos of Two Coupled Logistic Maps. Preprint (1986).

    Google Scholar 

  2. Beau, W., W.H. Hehl, W. Metzler: Computerbilder zur Analyse chaoserzeugender Abbildungen. Informatik Forsch. Entw. 2 (1987), 122–130.

    Google Scholar 

  3. Collet, P. and J.-P. Eckmann: Iterated Maps on the Interval as Dynamical Systems. A. Jaffe and D. Ruelle (eds.). Birkhäuser, Basel/Boston/Stuttgart 1980.

    Google Scholar 

  4. Cvitanovic, P. (ed.): Universality in Chaos. Adam Hilger Ltd., Bristol 1983.

    Google Scholar 

  5. Feigenbaum, M.: The Universal Metric Properties of Nonlinear Transformations. J. Stat. Phys. 21 (1979), 669–706.

    Article  MathSciNet  MATH  Google Scholar 

  6. Feit, S.D.: Characteristic Exponents and Strange Attractors. Commun, math. Phys. 61 (1978), 249.

    Article  MathSciNet  MATH  Google Scholar 

  7. Guckenheimer, J., P. Holmes: Nonlinear Oscillations, Dynamical Systems and Birfurcations of Vector Fields. Springer, New York/Berlin/Heidelberg/Tokyo 19862.

    Google Scholar 

  8. Haken, H. (ed.): Evolution of Order and Chaos in Physics, Chemistry, and Biology. Springer, Berlin 1982.

    MATH  Google Scholar 

  9. Hogg, T. and B.A. Huberman: Generic Behavior of Coupled Oscillators, Phys. Rev. A 29 (1984), 275.

    MathSciNet  Google Scholar 

  10. Kaneko, K.: Transition from Torus to Chaos Accompanied by Frequency Lockings with Symmetry Breaking. Prog. Theor. Phys. 69 (1983), 1427.

    Article  MATH  Google Scholar 

  11. Li, T.Y. and J.A. Yorke: Period Three Implies Chaos. Amer. Math. Monthly (1975), 958–992.

    Google Scholar 

  12. Mandelbrot, M.S.: The Fractal Geometry of Nature. Freeman, San Francisco 1982.

    MATH  Google Scholar 

  13. May, R.B.: Simple Mathematical Models with very Complicated Dynamics. Nature 261 (1976), 459–467.

    Article  Google Scholar 

  14. Metzler, W., W. Beau, W. Frees, A. Überla: Symmetry and Self-similarity with Coupled Logistic Maps. Z. Naturforsch. 42a (1987), 310–318.

    Google Scholar 

  15. Metzler, W.: Chaos und Fraktale bei zwei gekoppelten nichtlinearen Modelloszillatoren. PdN Physik 7/36 (1987), 23–29.

    Google Scholar 

  16. Metzler, W., W. Beau, A. Überla: A Route to Chaos. Computergraphics Film. Inst. f.d. Wiss. Film, C 1641, Göttingen 1987.

    Google Scholar 

  17. Peitgen, H.O., P.H. Richter: The Beauty of Fractals. Springer, Berlin/Heidelberg/New York/Tokyo 1986.

    Book  MATH  Google Scholar 

  18. Waller, H. and Kapral, R.: Spatial and Temporal Structure in Systems of Coupled Nonlinear Oscillators. Phys. Rev. A 30 (1984), 2047.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Kassel, P.O. Box 101380, D-3500, Kassel, Germany

    Wolfgang Metzler

Authors
  1. Wolfgang Metzler
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Editor information

Editors and Affiliations

  1. Akademie der Wissenschaften der DDR, Zentralinstitut für Kybernetik und Informationsprozesse, Berlin, GDR

    Achim Sydow

  2. Computer Engineering Division, National Technical University of Athens, Athens, Greece

    Spyros G. Tzafestas

  3. Department of Computer Science, Rutgers University, New Brunswick, NJ, USA

    Robert Vichnevetsky

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© 1988 Akademie-Verlag Berlin

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Metzler, W. (1988). A Route to Chaos. In: Sydow, A., Tzafestas, S.G., Vichnevetsky, R. (eds) Systems Analysis and Simulation I. Advances in Simulation, vol 1. Springer, New York, NY. https://doi.org/10.1007/978-1-4684-6389-7_4

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  • DOI: https://doi.org/10.1007/978-1-4684-6389-7_4

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-0-387-97091-2

  • Online ISBN: 978-1-4684-6389-7

  • eBook Packages: Springer Book Archive

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