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Continuous Chaos

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Synergetics

Part of the book series: Springer Series in Synergetics ((SSSYN,volume 2))

Abstract

While it is known that 3-variable continuous dynamical systems are ‘infinitely’ richer in their behavioral capabilities than two-variable ones (see, e.g., (1)), a systematic attempt at finding the simplest prototypes has yet to be made. Hereby a machinery for ‘composing’ higher-dimensional systems out of lower-dimensional subsystems such that the overall behavior remains predictable will be helpful. Liénard’s ‘decomposition’ method (into slow submanifolds and fast foliations, see (2)) can be ‘turned around’ (3) for this purpose.

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References

  1. Andronov, A.A., Leontovich, E.A., Gordon, I.I., and Maier, A.G., Theory of Bifurcations of Dynamic Systems on a Plane, Wiley: New York 1972. First Russian ed. 1967. See p. XXIII.

    Google Scholar 

  2. Zeeman, E.C., In: Towards a Theoretical Biology, Vol. 4, pp. 8–67 (C.H. Waddington, ed.), University Press: Edinburgh 1972.

    Google Scholar 

  3. Rössler, O.E., Lecture Notes in Biomathematics 4 (Springer-Verlag), 546 (1974).

    Google Scholar 

  4. Thom, R., In: Towards a Theoretical Biology (C.H. Waddington, ed.), Vol. 3, pp. 89–116, Aldine Publ. Comp.: Chicago 1970.

    Google Scholar 

  5. Rössler, O.E. and Denzel, B., Two-variable Stochastic Flip-flop in Abstract Kinetics (in preparation).

    Google Scholar 

  6. Khaikin, S.E., Zh. Prikl. Fiz. 7 (6), 21 (1930).

    Google Scholar 

  7. Andronov, A.A., Khaikin, S.E., and Vitt, A.A., Theory of Oscillators, p. 725, Pergamon: New York 1966. (First Russian ed. 1937.)

    MATH  Google Scholar 

  8. Rössler, O.E. and Hoffmann, D., 4th Int. Biophysics Congress, Moscow 1972 (Abstracts Vol. 4, p. 49).

    Google Scholar 

  9. Rössler, O.E., Z. Naturforsch. 31 a, 259 (1976).

    ADS  Google Scholar 

  10. Li, T.Y. and Yorke, J.A., Amer. Math. Monthly 82, 985 (1975).

    Article  MathSciNet  MATH  Google Scholar 

  11. Julia, G., J. Math. Pur. Appl., Série 8, 1 (1918).

    Google Scholar 

  12. Ulam, S.M., In: Mathematical Models in Physical Sciences (S. Drobot, ed.), pp. 85–95, Prentice Hall: Englewood Cliffs 1963.

    Google Scholar 

  13. Lorenz, E.N., J. Atmos. Sci. 20, 130 (1963).

    Article  ADS  Google Scholar 

  14. Sharkovskii, A.N., Ukranian Math. Jour. 20, 136 (1968).

    Google Scholar 

  15. Rössler, O.E., Bull. Math. Biol. 39, 275 (1977).

    MATH  Google Scholar 

  16. Rössler, O.E., Biophys. J. 17, 281a (1977). (Abstract.)

    Article  Google Scholar 

  17. Hull, T.E. and Dobell, A.R., Soc. Ind. Appl. Math. 4, 230 (1962).

    MathSciNet  MATH  Google Scholar 

  18. Rössler, O.E., Z. Naturforsch. 31 a, 1664 (1976).

    ADS  Google Scholar 

  19. Guckenheimer, J., In: The Hopf Bifurcation (J.E. Marsden and M. McCracken, eds.), pp. 368–381, Springer-Verlag: New York 1976.

    Chapter  Google Scholar 

  20. Williams, R.F., The Structure of Lorenz Attractors (preprint).

    Google Scholar 

  21. Ruelle, D. and Takens, F., Commun. Math. Phys. 20, 167 (1971).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  22. Guckenheimer, J., Structural Stability of Lorenz Attractors. (Preprint.)

    Google Scholar 

  23. Rössler, O.E., Syncope Implies Chaos in Walking-stick Maps, Z. Naturforsch, a (in press).

    Google Scholar 

  24. Smale, S., Bull. Amer. Math. Soc. 73, 747 (1967).

    Article  MathSciNet  MATH  Google Scholar 

  25. Poincaré, H., Les Méthodes Nouvelles de la Mécanique Céleste, Vols. 1–3, Paris 1899; reprint Dover: New York 1957.

    MATH  Google Scholar 

  26. Birkhoff, G.D., Acta Math. 50, 359 (1927).

    Article  MathSciNet  MATH  Google Scholar 

  27. Gumowski, I., R.A.I.R.O Automatique 10, 7 (1976).

    MathSciNet  Google Scholar 

  28. Rössler, O.E., Phys. Lett. 57 A, 397 (1976)

    Google Scholar 

  29. Rössler, O.E., Phys. Lett. 60 A, 392 (1977).

    MathSciNet  Google Scholar 

  30. Cook, A.F. and Roberts, P.H., Proc. Camb. Phil. Soc. 68, 547 (1970).

    Article  ADS  Google Scholar 

  31. Malkus, W.V.R., EOS. Trans. Am. Geophys. Union 53, 617 (1972).

    Google Scholar 

  32. Robbins, K.A., A New Approach to Subcritical Instability and Turbulent Transition in a Simple Dynamo, Proc. Camb. Phil. Soc. (in press).

    Google Scholar 

  33. Moore, D.W. and Spiegel, E.A., Astrophys. J. 143, 871 (1966).

    Article  MathSciNet  ADS  Google Scholar 

  34. Rössler, O.E., Z. Naturforsch. 32 a, 299 (1 977).

    ADS  Google Scholar 

  35. Smale, S., J. Differential Geometry 7, 193 (1972).

    MathSciNet  MATH  Google Scholar 

  36. Rössler, O.E. and Wegmann, K., Chaos in the Zhabotinskii Reaction (preprint).

    Google Scholar 

  37. Haken, H., Synergetics, Springer-Verlag: New York 1977.

    Google Scholar 

  38. Joseph, D.D., Stability of Fluid Motions, Vols. I and II, Springer-Verlag: New York 1976.

    Google Scholar 

  39. Haken, H., Phys. Lett. 53 A, 77 (1975).

    Google Scholar 

  40. Graham, R., Phys. Lett. 38 A, 440 (1976).

    Google Scholar 

  41. Kuramoto, Y. and Yamada, T., Progr. Theor. Phys. 56, 679 (1976).

    Article  MathSciNet  ADS  Google Scholar 

  42. Rössler, O.E., Z. Naturforsch. 31 a, 1168 (1976).

    ADS  Google Scholar 

  43. Rössler, O.E., Chemical Turbulence - A Synopsis, In: Synergetics (H. Haken, ed.), Lecture Notes in Physics (in press).

    Google Scholar 

  44. Kuramoto, Y., Chemical Turbulence and Chemical Waves, In: Synergetics (H. Haken, ed.), Lecture Notes in Physics (in press).

    Google Scholar 

  45. Mackey, M., Biophys. J. 17, 281a (1977). (Abstract.)

    Article  Google Scholar 

  46. May, R.M., Nature 261, 459 (1976).

    Article  ADS  Google Scholar 

  47. Guckenheimer, J., Oster, G., and Ipatchki, A., The Dynamics of Density Dependent Population Models. Theoret. Popul. Biol. (in press).

    Google Scholar 

  48. Newhouse, S. and Palis, J., Astérisque 31, 44 (1976).

    MATH  Google Scholar 

  49. Moser, J., Stable and Random Motions in Dynamical Systems with Special Emphasis on Celestrial Mechanics, Princeton University Press: Princeton 1973.

    Google Scholar 

  50. Takens, F., Lecture Notes in Mathematics 535, 237 (1976). Springer-Verlag.

    Article  MathSciNet  Google Scholar 

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Authors
  1. O. E. Rössler
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Editors and Affiliations

  1. Institut für Theoretische Physik, der Universität Stuttgart, Pfaffenwaldring 57/IV, D-7000, Stuttgart 80, Fed. Rep. of Germany

    Hermann Haken

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© 1977 Springer-Verlag Berlin Heidelberg

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Rössler, O.E. (1977). Continuous Chaos. In: Haken, H. (eds) Synergetics. Springer Series in Synergetics, vol 2. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-66784-8_17

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  • DOI: https://doi.org/10.1007/978-3-642-66784-8_17

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-66786-2

  • Online ISBN: 978-3-642-66784-8

  • eBook Packages: Springer Book Archive

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