Synergetics pp 174-183 | Cite as

Chemical Turbulence A Synopsis

  • O. E. Rössler
Conference paper
Part of the Springer Series in Synergetics book series (SSSYN, volume 2)


‘Tumbling’ is a ubiquitous behavioral possibility in natural systems. The two oldest physical examples are a rippling flow of water, on the one hand, and the space-time behavior of three gravitating masses, on the other hand. A proof that stricty nonperiodic behavior is possible in the last-mentioned case was given by POINCARE [1] (who detected a ‘homoclinic point’ in a cross-section through the trajectorial flow). Later LORENZ [2] devised his well-known reduced equation for turbulent NAVIER-STOKES flows, which also possesses nonperiodic trajectories.


Excitable Medium Stable Steady State Limit Cycle Oscillator Homoclinic Point Trajectorial Flow 
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  1. 1.
    H. Poincaré, Les Méthodes Nouvelles de la Méchanique Céleste, Vols. 1–3. Paris 1899. Reprint Dover, New York, 1957.Google Scholar
  2. 2.
    E.N. Lorenz, Deterministic Nonperiodic Flow. J. Atmos. Sci. 20, 130 (1963).ADSCrossRefGoogle Scholar
  3. 3.
    O.E. Rössler, Chaotic Behavior in Simple Reaction Systems. Z. Naturforsch. 31 a, 259 (1976).ADSGoogle Scholar
  4. 4.
    O.E. Rössler, Chaos in Abstract Kinetics: Two Prototypes. Bull. Math. Biol. 39, 275 (1977).zbMATHGoogle Scholar
  5. 5.
    O.E. Rössler, Different Types of Chaos in Two Simple Differential Equations. Z. Naturforsch. 31 a, 1664 (1976).ADSGoogle Scholar
  6. 6.
    O.E. Rössler, Toroidal Oscillations in a 3-Variable Abstract Reaction System. Z. Naturforsch, a (in press).Google Scholar
  7. 7.
    O.E. Rössler, Chaos in a Modified Danziger-Elmergreen Equation (in preparation).Google Scholar
  8. 8.
    T.Y. li and J.A. Yorke, Period Three Implies Chaos. Amer. Math. Monthly 82, 985 (1975).MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    S. Smale, Differentiable Dynamical Systems. Bull. Amer. Math. Soc. 73, 747 (1967).MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    D. Ruelle and F. Takens, On the Nature of Turbulence. Commun. Math. Phys. 20, 167 (1971).MathSciNetADSzbMATHCrossRefGoogle Scholar
  11. 11.
    O.E. Rössler, Chemical Turbulence: Chaos in a Simple Reaction-Diffusion System. Z. Naturforsch. 31 a, 1168 (1976).ADSGoogle Scholar
  12. 12.
    A.M. Turing, The Chemical Basis of Morphogenesis. Phil. Trans. Roy. Soc. London B 237, 37 (1952).ADSCrossRefGoogle Scholar
  13. 13.
    O.E. Rössler and F.F. Seelig, A Rashevsky-Turing system as a Two-cellular Flip-flop. Z. Naturforsch. 27 b, 1444 (1972).Google Scholar
  14. 14.
    O.E. Rössler, A Synthetic Approach to Exotic Kinetics, With Examples. Lecture Notes in Biomathematics 4, 546 (1974).Google Scholar
  15. 15.
    O.E. Rössler, Basic Circuits of Fluid Automata and Relaxation Systems (in German). Z. Naturforsch. 21 b, 333 (1 972).Google Scholar
  16. 16.
    A. Gierer and H. Meinhardt, A Theory of Biological Pattern Formation. Kybernetik 12, 30 (1972).CrossRefGoogle Scholar
  17. 17.
    I. Prigogine and G. Nicolis, On Symmetry-breaking Instabilities in Dissipative Systems. J. Chem. Phys. 46, 3542 (1967).ADSCrossRefGoogle Scholar
  18. 18.
    Y. Kuramoto and T. Yamada, Turbulent State in Chemical Reactions. Progr. Theor. Phys. 55, 679 (1976).MathSciNetADSCrossRefGoogle Scholar
  19. 19.
    A.T. Winfree, Spatial and Temporal Organization in the Zhabotinsky Reaction. In: Advances in Biological and Medical Physics (to appear).Google Scholar
  20. 20.
    A.T. Winfree, Scroll-shaped Waves of Chemical Activity in Three Dimensions. Science (Wash.) 181, 937 (1973).ADSCrossRefGoogle Scholar
  21. 21.
    A.T. Winfree, Personal Communication 1975.Google Scholar
  22. 22.
    O.E. Rössler and K. Wegmann, Chaos in the Zhabotinsky Reaction (in preparation).Google Scholar
  23. 23.
    O.E. Rössler and C. Kahlert, The Winfree Effect: Meandering in a 2-Variable 2-Dimensional Excitable Medium (in preparation).Google Scholar
  24. 24.
    H.R. Karfunkel and F.F. Seelig, Excitable Chemical Reaction Systems. I. Definition of Excitability and Simulation of Model Systems. J. Math. Biol. 2, 123 (1975).zbMATHCrossRefGoogle Scholar
  25. 25.
    P. Ortoleva and J. Ross, Theory of Propagation of Discontinuities in Kinetic Systems with Multiple Time Scales: Fronts, Front Multiplicities, and Pulses. J. Chem. Phys. 63, 3398 (1975).ADSCrossRefGoogle Scholar
  26. 26.
    O.E. Rössler, Chemical Automata in Homogeneous and Reaction-Diffusion Kinetics. Lecture Notes in Biomathematics 4, 399 (1974).Google Scholar
  27. 27.
    O.E. Rössler, Two-Variable Excitable Morphogenesis (in preparation).Google Scholar
  28. 28.
    J.L. Kaplan and J.A. Yorke, Preturbulence in a Regime Observed in a Fluid Model of Lorenz. Preprint 1977.Google Scholar
  29. 29.
    J. Cowan, Personal Communication 1977.Google Scholar
  30. 30.
    H.D. Landahl, A Mathematical Model for First Degree Blocks and the Wenckebach Phenomenon. Bull. Biophysics 33, 27 (1971).CrossRefGoogle Scholar
  31. 31.
    D.J. Graves, N. Yang and R.A. Tipton, Unusual Modification of Immobilized Enzyme Kinetics Caused by Diffusional Resistance. Preprint 1976.Google Scholar
  32. 32.
    N.S. Goel, S.C. Maitra and E.W. Montroll, Nonlinear Models of Interacting Populations. Academic Press, New York, 1971.Google Scholar
  33. 33.
    E.H. Kerner, A Statistical Mechanics of Interacting Biological Species. Bull. Math. Biophysics 19, 1 21 (1957).MathSciNetCrossRefGoogle Scholar
  34. 34.
    H. Haken, Analogy Between Higher Instabilities in Fluids and Lasers. Phys. Lett. 53 A, 77 (1975).Google Scholar

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© Springer-Verlag Berlin Heidelberg 1977

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  • O. E. Rössler

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