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Part of the book series: Springer Series in Synergetics ((SSSYN,volume 27))

Abstract

RUELLE [1] suggested more than a decade ago that since nonequilibrium chemical reactions are described by coupled nonlinear differential equations, for some conditions they might exhibit nonperiodic behavior. The nonperiodic behavior that arises from the nonlinear nature of a system rather than from stochastic driving forces is now called chaos, a term that we will define more carefully later.

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References

  1. D. Ruelle, Trans. N.Y. Acad. Sci. II, 35, 66 (1973).

    Google Scholar 

  2. L.F. Olsen and H. Degn, Nature 267, 177 (1977).

    Article  ADS  Google Scholar 

  3. T. Li and J. Yorke, Am. Math. Monthly 82, 985 (1975). This paper was the first to use the term “chaos” with the meaning that it has in the title of the present paper.

    Article  MathSciNet  MATH  Google Scholar 

  4. R.A. Schmitz, K.R. Graziani, J.L. Hudson, J. Chem. Phys. 67, 3040 (1977).

    Article  ADS  Google Scholar 

  5. O. Rössler, K. Wegmann, Nature 271, 89 (1978);

    Article  Google Scholar 

  6. K. Wegmann, O. Rössler, Z. Naturforsch. 33a, 1179 (1978).

    ADS  Google Scholar 

  7. See also O.E. Rössler, “Chemical Turbulence,” in Synergetics, A Workshop, ed. by H. Haken (Springer, 1977), p. 174.

    Chapter  Google Scholar 

  8. C. Vidal, J.C. Roux, A. Rossi, S. Bachelart, C.R. Acad. Sci. Paris 289C, 73 (1979).

    Google Scholar 

  9. J.L. Hudson, M. Hart, D. Marinko, J. Chem. Phys. 71, 1601 (1979).

    Article  ADS  Google Scholar 

  10. P.G. Sorenson, Ann. N.Y Acad. Sci. 316, 667 (1979).

    Article  ADS  Google Scholar 

  11. H. Nagashima, J. Phys. Soc. Japan 49, 2427 (1980).

    Article  Google Scholar 

  12. C. Vidal and A. Pacault, eds., Nonlinear Phenomena in Chemical Dynamics (Springer, Berlin, 1981).

    Google Scholar 

  13. See ref. 10: D. Ruelle, p. 30; O.E. Rössler, p. 79.

    Google Scholar 

  14. See ref. 10: J.C. Roux and H.L. Swinney, p. 33; J.L. Hudson, J. Mankin, J. McCullough, P. Lamba, p. 44; and C. Vidal, p. 49.

    Google Scholar 

  15. C. Vidal, J.C. Roux, S. Bachelart, A. Rossi, Ann. N.Y. Acad. Sci. 357, 377 (1980).

    Article  ADS  Google Scholar 

  16. J.C. Roux, A. Rossi, S. Bachelart, C. Vidal, Phys. Lett. 77A, 391 (1980).

    MathSciNet  ADS  Google Scholar 

  17. J.C. Roux, A. Rossi, S. Bachelart, C. Vidal, Physica 2D, 395 (1981).

    MathSciNet  ADS  Google Scholar 

  18. Y. Pomeau, J.C. Roux, A. Rossi, S. Bachelart, C. Vidal, J. Phys. Lett. 42, L271 (1981).

    Article  Google Scholar 

  19. C. Vidal, S. Bachelart, A. Rossi, J. Phys. (Paris) 43, 7 (1982).

    Article  Google Scholar 

  20. J.C. Roux: Physica 7D, 57 (1983).

    MathSciNet  ADS  Google Scholar 

  21. J.L. Hudson and J.C. Mankin, J. Chem. Phys. 74, 6171 (1981).

    Article  ADS  Google Scholar 

  22. J.L. Hudson, J. Mankin, J. McCullough, P. Lamba, in ref. 10, p. 44.

    Google Scholar 

  23. J.S. Turner, J.C. Roux, W.D. McCormick, H.L. Swinney, Phys. Lett. 85A, 9 (1981).

    ADS  Google Scholar 

  24. J.C. Roux and H.L. Swinney, in ref. 10, p. 33.

    Google Scholar 

  25. J.C. Roux, J.S. Turner, W.D. McCormick, H.L. Swinney, in Nonlinear Problems: Present and Future, A.R. Bishop, D.K. Campbell, B. Nicolaenko, eds. (North-Holland, Amsterdam, 1982), p. 409.

    Chapter  Google Scholar 

  26. R.H. Simoyi, A. Wolf, H.L. Swinney, Phys. Rev. Lett. 49, 245 (1982).

    Article  MathSciNet  ADS  Google Scholar 

  27. J.C. Roux, R.H. Simoyi, H.L. Swinney, Physica 8D, 257 (1983).

    MathSciNet  ADS  Google Scholar 

  28. K. Showalter, R.M. Noyes, K. Bar-Eli, J. Chem. Phys. 69, 2514 (1978).

    Article  ADS  Google Scholar 

  29. R.M. Noyes, in Stochastic Phenomena and Chaotic Behavior in Complex Systems, ed. by P. Schuster (Springer, Berlin, 1984), p. 106;

    Chapter  Google Scholar 

  30. N. Ganapathisubramanian and R.M. Noyes, J. Chem. Phys. 76, 1770 (1982).

    Article  ADS  Google Scholar 

  31. I.B. Schwartz, Phys. Lett. 102A, 25 (1984).

    ADS  Google Scholar 

  32. L.F. Olsen, in Stochastic Phenomena and Chaotic Behavior in Complex Systems, ed. by P. Schuster (Springer, Berlin, 1984), p. 116.

    Chapter  Google Scholar 

  33. I.R. Epstein, Physica 7D, 47 (1983); see also the paper by Epstein in ref. 46.

    Google Scholar 

  34. D. Ruelle, private communication.

    Google Scholar 

  35. N.H. Packard, J.P. Crutchfield, J.D. Farmer, R.S. Shaw, Phys. Rev. Lett. 45, 712 (1980).

    Article  ADS  Google Scholar 

  36. F. Takens, DUNG Lecture Notes in Mathematics, 898, ed. by D.A. Rand and L.S. Young (Springer, Berlin, 1981), p. 366.

    Google Scholar 

  37. H.S. Greenside, G. Ahlers, P.C. Hohenberg, R.W. Walden, Physica 5D, 322 (1982).

    MathSciNet  ADS  Google Scholar 

  38. D. Farmer, E. Ott, J. Yorke, Physica 7D, 153 (1983).

    MathSciNet  ADS  Google Scholar 

  39. A. Wolf and J. Swift, in Statistical Physics and Chaos in Fusion Plasmas, ed. by W. Horton and L. Reichl (Wiley, New York, 1984), p. 111.

    Google Scholar 

  40. A. Wolf, J. Swift, H.L. Swinney, J. Vastano, “Determining Lyapunov exponents from a time series,” submitted to Physica D.

    Google Scholar 

  41. D. Ruelle, Ann. N.Y. Acad. Sci. 317, 408 (1978).

    Google Scholar 

  42. P. Grassberger and I. Procaccia, Phys. Rev. Lett. 50, 346 (1983).

    Article  MathSciNet  ADS  Google Scholar 

  43. B. B. Mandelbrot, The Fractal Geometry of Nature (Freeman, San Francisco, 1982). See the discussion of chaotic (“fractal”) attractors, pp. 193–198.

    MATH  Google Scholar 

  44. J.D. Farmer, E. Ott, J. Yorke, Physica 7D, 153 (1983).

    MathSciNet  ADS  Google Scholar 

  45. A. Brandstater, J. Swift, H.L. Swinney, A. Wolf, J.D. Farmer, E. Jen, J.P. Crutchfield, Phys. Rev. Lett. 51, 1442 (1983).

    Article  MathSciNet  ADS  Google Scholar 

  46. A. Brandstater, J. Swift, H.L. Swinney, A. Wolf, in Turbulence and Chaotic Phenomena in Fluids, ed. by T. Tatsumi (North-Holland, Amsterdam, 1984);

    Google Scholar 

  47. A. Brandstater and H.L. Swinney, in Fluctuations and Sensitivity in Nonequilibrium Systems, ed. by W. Horsthemke and D. Kondepudi (Springer, Berlin, 1984).

    Google Scholar 

  48. K. Coffman, W.D. McCormick, J.C. Roux, R. Simoyi, H.L. Swinney, to be published.

    Google Scholar 

  49. J. Maselko and H.L. Swinney, Physica Scripta, to appear (1984).

    Google Scholar 

  50. Nonequilibrium Dynamics in Chemical Systems, ed. by C. Vidal and A. Pacault (Springer, Berlin, 1984) (this volume).

    Google Scholar 

  51. K. Coffman, W.D. McCormick, H.L. Swinney, J.C. Roux, in ref. 46. Also, K. Coffman, W.D. McCormick, J.C. Roux, R. Simoyi, H.L. Swinney, to be published.

    Google Scholar 

  52. M.J. Feigenbaum, J. Stat. Phys. 19, 25 (1978).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  53. M.J. Feigenbaum, Physica 7D, 16 (1983).

    MathSciNet  ADS  Google Scholar 

  54. P. Collet and J.P. Eckmann, Iterated Maps of the Interval as Dynamical Systems (Birkhauser, Boston, 1980).

    Google Scholar 

  55. Y. Pomeau and P. Manneville, Commun. Math. Phys. 74, 189 (1980).

    Article  MathSciNet  ADS  Google Scholar 

  56. J.C. Roux, A. Rossi, in ref. 46.

    Google Scholar 

  57. F. Argoul, P. Richetti, A. Arneodo, in ref. 46.

    Google Scholar 

  58. I. Prigogine and P. Lefever, J. Chem. Phys. 48, 1695 (1968).

    Article  ADS  Google Scholar 

  59. F. Schlögl, Z. Phys. 248, 446 (1971).

    Article  ADS  Google Scholar 

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

    Google Scholar 

  61. O.E. Rössler, Z. Naturforsch. Bull. Math. Biol, 39, 275 (1977).

    MATH  Google Scholar 

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

    Article  ADS  Google Scholar 

  63. O. Decroly, A. Goldbeter, Proc. Natl. Acad. Sci. (USA) 79, 6917 (1982).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  64. L.F. Olsen, Phys. Lett. 94A, 454 (1982).

    ADS  Google Scholar 

  65. R.J. Field, E. Körös, and R.M. Noyes, J. Am. Chem. Soc. 94, 8649 (1972).

    Article  Google Scholar 

  66. R.J. Field and R.M. Noyes, J. Chem. Phys. 60, 1877 (1974).

    Article  ADS  Google Scholar 

  67. J. Rinzel, I.B. Schwartz, J. Chem. Phys. 80, 5610 (1984).

    Article  ADS  Google Scholar 

  68. J.J. Tyson, J. Math. Biol. 5, 351 (1978).

    MathSciNet  MATH  Google Scholar 

  69. K. Tomita, I. Tsuda, Phys. Lett. 71A, 489 (1979).

    MathSciNet  ADS  Google Scholar 

  70. A.S. Pikovsky, Phys. Lett. 85A, 13 (1980)

    MathSciNet  ADS  Google Scholar 

  71. A.S. Pikovsky, M.I. Rabinovich, Physica 2D, 8 (1981).

    MathSciNet  ADS  Google Scholar 

  72. C. Lobry and R. Lozi, in ref. 10, p. 67.

    Google Scholar 

  73. J. Turner, Discussion Meeting, Kinetics of Physicochemical Oscillations (Aachen, September, 1979).

    Google Scholar 

  74. J. Turner, in Self-Organization and Dissipative Structures, ed. by W.C. Schieve and P. Allen (University of Texas Press, Austin, 1982), p. 41.

    Google Scholar 

  75. J. Ringland and J.S. Turner, “One-dimensional behavior in a model of the Belousov-Zhabotinskii reaction,” Phys. Lett., in press (1984).

    Google Scholar 

  76. D. Lindberg and J.S. Turner, to be published.

    Google Scholar 

  77. Z. Noszticzius, H. Farkas, and Z.A. Schelly, J. Chem. Phys. 80, 6062(1984).

    MathSciNet  Google Scholar 

  78. R.M. Noyes, J. Chem. Phys. 80, 6071 (1984).

    Article  MathSciNet  ADS  Google Scholar 

  79. J.J. Tyson, J. Chem. Phys. 80, 6079 (1984).

    Article  MathSciNet  ADS  Google Scholar 

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Swinney, H.L., Roux, J.C. (1984). Chemical Chaos. In: Vidal, C., Pacault, A. (eds) Non-Equilibrium Dynamics in Chemical Systems. Springer Series in Synergetics, vol 27. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-70196-2_20

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  • DOI: https://doi.org/10.1007/978-3-642-70196-2_20

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