Skip to main content

Part of the book series: Springer Series in Synergetics ((SSSYN,volume 4))

Abstract

‘Chaos’ is the canonic translation into Greek of the Hebrew term ‘tohu-wa-bohu’ found in the first chapter of the Bible. ‘Wa’ means ‘and,’ and ‘bohu’ certainly means the same thing as ‘tohu.’ However, since the word appears only once in the Bible and there is no continuous oral tradition, one can only guess what ‘tohu’ means. A possible English analogue is ‘topsy(-and-)turvy.’ A mathematical redefinition thus seems admissible.

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 39.99
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

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. Li, T.Y. and J. Yorke (1975). Period three implies chaos. Amer. Math. Monthly 82, 985–992

    Article  MathSciNet  MATH  Google Scholar 

  2. May, R.M. (1974). Biological populations with nonoverlapping generations: Stable points, limit cycles, and chaos. Science 186, 645–647

    Article  ADS  Google Scholar 

  3. Hoppensteadt, F.C. and J.M. Hyman (1977). Periodic solutions of a logistic difference equation. SIAM J. Appl. Math. 32, 73–81

    Article  MathSciNet  MATH  Google Scholar 

  4. Julia, G. (1918). Mémoire sur l’itération des fonctions rationelles. J. Math. Pur. Appl., Série 8, 1, 1–18

    Google Scholar 

  5. Smale, S. and R.F. Williams (1976). The qualitative analysis of a difference equation of population growth. J. Math. Biol. 3, 1–5

    Article  MathSciNet  Google Scholar 

  6. Myrberg, P.J. (1963). Iteration of the real polynomials of second degree III (in German). Ann. Acad. Sci. Fenn. Ser. A, 336/3, 1–18

    Google Scholar 

  7. Gumowski, I. and C. Mira (1969). Sensitivity problems related to certain bifurcations in the nonlinear recurrence relations. Automatica, the Journal of IFAC 5, 303–317

    Article  MathSciNet  MATH  Google Scholar 

  8. Grossmann, S. and S. Thomae (1977). Invariant distribution and stationary correlation functions of one-dimensional discrete processes. Z. Naturforsch. 32a, 1353–1363

    MathSciNet  ADS  Google Scholar 

  9. Sharkovsky, A.N. (1964). Coexistence of the cycles of a continuous mapping of the line into itself. Ukrain. Mat. Z. 16, 61–71

    MathSciNet  Google Scholar 

  10. Guckenheimer, J. (1979). Dynamical systems. In: Lectures in Applied Mathematics, Vol. 17 (F.C. Hoppensteadt, ed.), Providence, R.I.: Amer. Math. Soc.

    Google Scholar 

  11. Hill, G.W. (1878). Researches in the lunar theory. Amer. J. Math, 1, 5–26

    Article  MathSciNet  Google Scholar 

  12. Poincaré, H. (1890). Sur le problème des trois corps et les équations de la dynamique. Acta Math. 13, 1–271

    MATH  Google Scholar 

  13. Poincaré, H. (1899). Les Méthodes Nouvelles de la Mécanique Céleste, Vol. III, Chapter 27; p. 387. Paris: Gauthier-Villars

    MATH  Google Scholar 

  14. Birkhoff, G.D. (1920). Recent advances in dynamics. Science 51, 51–55

    Article  ADS  Google Scholar 

  15. Hartman, P. (1964). Ordinary Differential Equations. New York: Wiley

    MATH  Google Scholar 

  16. Birkhoff, G.D. (1927). On the periodic motions of dynamical systems. Acta Math. 50, 359–379

    Article  MathSciNet  MATH  Google Scholar 

  17. Smale, S. (1967). Differentiable dynamical systems. Bull. Amer. Math. Soc. 73, 747–817

    Article  MathSciNet  MATH  Google Scholar 

  18. Hadamard, J. (1898). Les surfaces à curbures opposes et leurs lignes géodésiques. Journ. de Math. (5)4, 27–73

    Google Scholar 

  19. Anosov, D.V. and J.G. Sinai (1967). Some smooth ergodic systems. Uspekhi Mat. Nauk 22, 107–172

    MathSciNet  MATH  Google Scholar 

  20. Arnold, V.I. and A. Avez (1967). Théorie Ergodique des Systèmes Dynamiques. Paris: Gauthier-Villars

    Google Scholar 

  21. Moser, J. (1973). Stable and Random Motions in Dynamical Systems. Princeton, N.J.: Princeton Univ. Press

    MATH  Google Scholar 

  22. Gumowski, I. (1979). Monograph on Point Recurrences (forthcoming)

    Google Scholar 

  23. Levinson, N. (1949). A second order differential equation with singular solutions. Ann. Math. 50, 127–153

    Article  MathSciNet  Google Scholar 

  24. Cartwright, M.L. and J.E. Littlewood (1945). On nonlinear differential equations of the second order: I. The equation ÿ - k(l - y2)y+y = bλk cos(λt + α), k large. J. Lond. Math. Soc. 20, 180–189

    Article  MathSciNet  MATH  Google Scholar 

  25. Rössler, O.E. (1977). Quasiperiodic oscillation in an abstract reaction system. Biophysical J. 17, 281a (abstract)

    Article  Google Scholar 

  26. Rössler, O.E. (1979). Chaotic oscillations: An example of hyperchaos. In: Lectures in Applied Mathematics, Vol. 17 (F.C. Hoppensteadt, ed.), Providence, R.I.: Amer. Math. Soc.

    Google Scholar 

  27. Lorenz, E.N. (1963). Deterministic nonperiodic flow. J. Atmos. Sci. 20, 130–141

    Article  ADS  Google Scholar 

  28. Guckenheimer, J. (1976). A strange, strange attractor. In: The Hopf Bifurcation and Its Applications (J.E. Marsden and M. McCracken, eds.), pp. 368–381, New York: Springer-Verlag

    Chapter  Google Scholar 

  29. Rössler, O.E. (1976). Different types of chaos in two simple differential equations. Z. Naturforsch. 31a, 1664–1670

    ADS  Google Scholar 

  30. Williams, R.F. (1978). The bifurcation space of the Lorenz attractor. In: Bifurcation Theory and Applications in Scientific Disciplines (O. Gurel and O.E. Rössler, eds.), Proc. N.Y. Acad. Sci. 316

    Google Scholar 

  31. Moore, D.W. and E.A. Spiegel (1966). A thermally excited nonlinear oscillator. Astrophys. J. 143, 871–887

    Article  MathSciNet  ADS  Google Scholar 

  32. Cook, A.F. and P.H. Roberts (1970). The Rikitake two-disc dynamo system. Proc. Camb. Phil. Soc. 68, 547–569

    Article  ADS  Google Scholar 

  33. Williams, R.F. (1974). Expanding attractors. Public. Math. de 1’Institut des Hautes Etudes Scientifiques 43, 169–203

    Article  Google Scholar 

  34. Rössler, O.E. (1976). Chaotic behavior in simple reaction systems. Z. Naturforsch. 31a, 259–264

    ADS  Google Scholar 

  35. Rössler, O.E. (1978). Continuous chaos-four prototype equations. In: Bifurcation Theory and Applications in Scientific Disciplines (O. Gurel and O.E. Rössler, eds.), Proc. N.Y. Acad. Sci. 316, 376–394

    Google Scholar 

  36. Mira, C. (1978). Dynamique complexe engendrée par une équation différentielle d’ordre 3. Proceedings “Equadiff 78” (R. Conti, G. Sestini, and G. Villari, eds.), Florence, May 24–30, 1978, pp. 25–37

    Google Scholar 

  37. Rössler, O.E. (1977). Chaos in abstract kinetics: Two prototypes. Bull. Math. Biol. 39, 275–289

    MATH  Google Scholar 

  38. Rössler, O.E. (1977). Continuous chaos. In: Synergetics-A Workshop (H. Haken, ed.), pp. 184–197. Heidelberg/New York: Springer-Verlag

    Google Scholar 

  39. Takens, F. (1976). Implicit differential equations: Some open problems. Springer Lecture Notes in Math. 535, 237–253

    Article  MathSciNet  Google Scholar 

  40. Rössler, O.E. (1978). Chaotic oscillations in a 3-variable quadratic mass action system. In: Proc. Int’l Symp. Math. Topics in Biology, Kyoto, Sept. 1978, pp. 131–135, Publ. Kyoto Research Institute for Math. Sci.

    Google Scholar 

  41. Rössler, O.E., R. Rössler, and H.D. Landahl (1978). Arrhythmia in a periodically forced excitable system. Sixth Int’l Biophysics Congress, Kyoto, Japan. Abstracts Vol. p. 296

    Google Scholar 

  42. Marsden, J.E. (1977). Attempts to relate the Navier-Stokes equations to turbulence. In: Turbulence Seminar (A. Chorin, J. Marsden, and S. Smale, orgs.), Springer Lecture Notes in Math. 615, 1–22.

    Google Scholar 

  43. Hénon, M. and Y. Pomeau (1976). Two strange attractors with a simple structure. In: Springer Lecture Notes in Math. 565, 29–68

    Article  Google Scholar 

  44. Rössler, O.E. (1977). Horseshoe-map chaos in the Lorenz equation. Phys. Lett. 60A, 392–394

    ADS  Google Scholar 

  45. Rössler, R., F. Götz, and O.E. Rössler (1979). Chaos in endocrinology. Biophys. J. 25(2), 216a

    Google Scholar 

  46. Rössler, O.E. (1978). Chaos and strange attractors in chemical kinetics. In: Synergetics-Far from Equilibrium (A. Pacault and C. Vidal, eds.), pp. 107–113, Springer-Verlag

    Google Scholar 

  47. Hénon, M. (1976). A two-dimensional mapping with a strange attractor. Comm. Math. Phys. 50, 69–78

    Article  MathSciNet  ADS  MATH  Google Scholar 

  48. Curry, J.H. (1979). A homoclinic point in the Hénon map. In: Lectures in Applied Mathematics, Vol. 17 (F.C. Hoppensteadt, ed.), Providence, R.I.: Amer. Math. Soc.

    Google Scholar 

  49. Mira, C. and I. Gumowski (1979). In preparation

    Google Scholar 

  50. Newhouse, S.E. (1970). Nondensity of Axiom A(a) on S2. In: Global Analysis, Summer Institute Berkeley 1968 (S. Chern and S. Smale, eds.), Providence, R.I., Amer. Math. Soc: Proc. Symp. Pure Math. 14, 191–202

    Google Scholar 

  51. Nitecki, Z. (1971). Differentiable Dynamics. Cambridge, Mass.: M.I.T. Press

    MATH  Google Scholar 

  52. Chillingworth, D.R.J. (1976). Differential Topology with a View to Applications. Research Notes in Mathematics 9, London: Pitman

    Google Scholar 

  53. Abraham, R. and J.E. Marsden (1978). Foundations of Mechanics. 2nd enlarged ed. Reading, Mass.: Benjamin/Cummings

    MATH  Google Scholar 

  54. Newhouse, S.E. (1974). Diffeomorphisms with infinitely many sinks. Topology 13, 9–18

    Article  MathSciNet  MATH  Google Scholar 

  55. Newhouse, S. and J. Palis (1976). Cycles and bifurcation theory. Astérisque 31, 44–140

    MATH  Google Scholar 

  56. Rössler, O.E. (1979). An equation for hyperchaos. Submitted to Phys. Lett. A.

    Google Scholar 

  57. Rössler, O.E., I. Gumowski and C. Mira (1979). In preparation

    Google Scholar 

  58. Kuramoto, Y. (1978). Diffusion-induced chaos in reaction systems. Progr. Theor. Phys. 64, Suppl.

    Google Scholar 

  59. Voss, R.F. and J. Clark (1975). “1/F” noise in music and speech. Nature 258, 317–318

    Article  ADS  Google Scholar 

  60. Thomas, H. (1978). Instabilities and fluctuations in systems far from equilibrium. In: Proc. 5th Int’l Conf. Noise in Physical Systems, Bad Nauheim, March 1978

    Google Scholar 

  61. Johannesma, P.I.M. (1979). In preparation

    Google Scholar 

  62. Franks, J.M. (1977). The dimensions of basic sets. J. Differential Geometry 12, 435–441

    MathSciNet  MATH  Google Scholar 

  63. Rössler, O.E. (1976). An equation for continuous chaos. Phys. Lett. 57A, 397–398

    ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1979 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Rössler, O.E. (1979). Chaos. In: Güttinger, W., Eikemeier, H. (eds) Structural Stability in Physics. Springer Series in Synergetics, vol 4. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-67363-4_26

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-67363-4_26

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-67365-8

  • Online ISBN: 978-3-642-67363-4

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics