Deterministic Chaos in Chemical Reactions

  • F. W. Schneider
  • A. F. Münster
Conference paper
Part of the Springer Series in Synergetics book series (SSSYN, volume 48)


We verify the existence of chaos in a chemical reaction, for which we chose the Belousov-Zhabotinsky reaction at long residence times above 30 min in an efficiently stirred reactor by the use of a number of measures: Fourier spectra, autocorrelation functions, attractor construction, Poincaré sections, one-dimensional maps, maximum Lyapunov exponents and correlation dimensions. Period doubling and “period three” oscillations provide indications of deterministic chaos in this system. On the other hand we show that the reported chaotic oscillations at short residence times (less than 10 min) are due to amplified statistical fluctuations in a region where Farey ordered periodic states exist. The excitability of a steady state (focus) close to a bifurcation is demonstrated experimentally by superimposing Gaussian distributed variations on the flow rate. Numerical simulations using the modified Oregonator model are in agreement with our fluctuation experiments at high flow rates. Thus chemical chaos exists in the BZ reaction at low flow rates only.


Hopf Bifurcation High Flow Rate Fourier Spectrum Chaotic Motion Malonic Acid 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Schmitz, R.A.; Graziani, K.R.; Hudson, J.L.; J. Chem. Phys., 1977, 67, 3040.ADSCrossRefGoogle Scholar
  2. [2]
    Hudson, J.L.; Hart, M.; Marinko, D.J.; J. Chem. Phys., 1979, 71, 1601.ADSCrossRefGoogle Scholar
  3. [3]
    Roux, J.C.; Rossi, A.; in “Non-Equilibrium Dynamics in Chemical Systems”, Vidal, C.; Pacault, A. Eds., Springer: Berlin 1984, p. 141.CrossRefGoogle Scholar
  4. [4]
    Argoul, F.; Arneodo, A.; Richetti, P.; Roux, J.C.; J. Chem. Phys., 1987, 86, 3325.ADSCrossRefMathSciNetGoogle Scholar
  5. [5]
    Swinney, H.L.; Roux, J.C.; in “Non-Equilibrium Dynamics in Chemical Systems”, Vidal, C.; Pacault, A. Eds., Springer: Berlin 1984, p. 124.CrossRefGoogle Scholar
  6. [6]
    Turner, J.S.; Roux, J.C.; McCormick, W.D.; Swinney, H.L.; Phys. Lett., 1981, 85A, 9.ADSCrossRefGoogle Scholar
  7. [7]
    Wegmann K., Rössler O.E., Z. Naturforsch. 330, 1978, 1179.Google Scholar
  8. [8]
    Roux J.C., Physica 7D, 1983, 57.ADSMathSciNetGoogle Scholar
  9. [9]
    Simoyi, R.H., Wolf A., Swinney, H.L.; Phys. Rev. Lett., 1982, 49, 245.ADSCrossRefMathSciNetGoogle Scholar
  10. [10]
    Coffman K.G., McCormick W.D., Noszticzius Z., Simoyi R.H., Swinney H.L., J. Chem. Phys., 1987, 86, 119.ADSCrossRefGoogle Scholar
  11. [11]
    Roux, J.C.; Rossi, A.; Bachelart, S.; Vidal, C.; Phys. Lett. A, 1980, 77, 391.ADSCrossRefMathSciNetGoogle Scholar
  12. [12]
    Rössler, O.E.; Wegmann, K.; Nature, 1978, 271, 89.CrossRefGoogle Scholar
  13. [13]
    Vidal, C.; Roux, J.C.; Rossi, A.; Bachelart, S.; Seances Acad. Sci. Ser. C, 1979, 289, 73.Google Scholar
  14. [14]
    Vidal, C.; Roux, J.C.; Bachelart, S.; Rossi, A.; Ann. N.Y. Acad. Sci., 1980, 357, 377.ADSCrossRefGoogle Scholar
  15. [15]
    Vidal, C.; Bachelart, S.; Rossi, A.; J. Phys. [Les Ulis, Fr.], 1982, 113, 7.CrossRefGoogle Scholar
  16. [16]
    Roux, J.C.; Turner, J.S.; McCormick, W.D.; Swinney, H.L.; in “Nonlinear Problems: Present and Future”, Bishop, A.R.; Campbell, D.K.; Nicolaenko, B. Eds., North-Holland, Amsterdam 1982, p. 409.Google Scholar
  17. [17]
    Agoul, F.; Arneodo, A.; Richetti, P.; Roux, J.C.; Swinney, H.L; Acc. Chem. Res., 1987, 20, 436.CrossRefGoogle Scholar
  18. [18]
    Noszticzius; Z., McCormick, W.D., Swinney, H.L.; J. Phys. Chem, 1987, 91, 5129.CrossRefGoogle Scholar
  19. [19]
    Hudson, J.L.; Mankin, J.C.; J. Chem. Phys, 1981, 74, 6171.ADSCrossRefGoogle Scholar
  20. [20]
    Olsen L.F.; Degn, H.; Nature, 1977, 267, 177.ADSCrossRefGoogle Scholar
  21. [21]
    Boiteux, A.; Goldbeter, A.; Hess, B.; Procl. Natl. Acad. Sci. USA, 1975, 72, 3829.ADSCrossRefGoogle Scholar
  22. [22]
    Markus, M.; Kuschmitz, D.; Hess, B.; FEBS Lett., 1984, 172, 235.CrossRefGoogle Scholar
  23. [23]
    Imbiehl, R.; Cox, M.P.; Ertl, G.; J. Chem. Phys., 1986, 84, 3519.ADSCrossRefGoogle Scholar
  24. [24]
    Jaeger, W.I.; Moller, K.; Plath, P.; Z. Naturforsch., 1981, A36, 1012.ADSGoogle Scholar
  25. [25]
    Wicke, E.; Onken, H.U.; personal communication.Google Scholar
  26. [26]
    Belousov, B.P.; Ref.Radiats.Med., 1958, Moscow: Medgiz, p. 145.Google Scholar
  27. [27]
    Schneider, F.W.; Ann. Rev. Phys. Chem., 1985, 36, 347.ADSCrossRefGoogle Scholar
  28. [28]
    Marek, M.; in “Temporal Order”, Rensing, L., Jaeger, N.I. Eds., Springer, Berlin 1985.Google Scholar
  29. [29]
    Dolnik, M.; Schreiber, I.; Marek, M.; Phys. Lett., 1984, A100, 316.ADSCrossRefGoogle Scholar
  30. [30]
    Dolnik, M.; Finkeova, J.; Schreiber, I.; Marek, M.; J. Chem. Phys., 1989, 93, 2764.CrossRefGoogle Scholar
  31. [31]
    Buchholz, F.; Freund, A.; Schneider, F.W.; in“Temporal Order”, Rensing, L.; Jaeger, N.J. Eds., Springer, Berlin 1895, p. 116.Google Scholar
  32. [32]
    Lamba, P.; Hudson, J.L.; Chemical Engeneering Science, 1985, 41, 1.Google Scholar
  33. [33]
    Freund, A.; Buchholz, F.; Schneider, F.W.; Ber. Bunsenges. Phys. Chem.; 1985, 89, 637.CrossRefGoogle Scholar
  34. [34]
    Schuster, H.G.; “Deterministic Chaos”, Second Edition, VCH Verlag, Weinheim 1988.Google Scholar
  35. [35]
    Bergé, P.; Pomeau, Y.; Vidal, C.; “Order within Chaos”, Wiley, New York 1986.zbMATHGoogle Scholar
  36. [36]
    Takens, F.; in “Lecture Notes in Mathematics”, 898, Springer, Berlin 1981.Google Scholar
  37. [37]
    Packard, N.H.; Crutchfield, J.M.; Farmer, J.D.; Shaw, R.S.; Phys. Rev. Lett., 1980, 45, 712.ADSCrossRefGoogle Scholar
  38. [38]
    King, G.; Jones, R.; Broomhead, D.S.; in “Proceedings on Chaos 1987”, Nuclear Physics B, 1987.Google Scholar
  39. [39]
    Benettin, G.; Galgani, L.; International School on Intrinsic Stochasticy in Plasmas, Cargee, France, 1979.Google Scholar
  40. [40]
    Benettin, G.; Galgani, L.; Giorgilli, A.; Strelcyn, J.M.; Meccanica, 1980, 15, 9.ADSCrossRefzbMATHGoogle Scholar
  41. [41]
    Grassberger, P.; Procaccia, I.; Phys. Rev. Lett., 1983, 50, 346.ADSCrossRefMathSciNetGoogle Scholar
  42. [42]
    Grassberger, P.; Procaccia, I.; Physica, 1983, 9D, 189.ADSMathSciNetGoogle Scholar
  43. [43]
    Bar-Eli, K.; Noyes, R.M.; J. Chem. Phys., 1988, 88, 3646.ADSCrossRefGoogle Scholar
  44. [44]
    Ben-Mizrachi, A.; Procaccia, I.; Phys. Rev., 1984, 29, 975.ADSCrossRefGoogle Scholar
  45. [45]
    Showalter, K.; Noyes, R.M.; Bar-Eli, K.; J. Chem. Phys., 1978, 69, 2514.ADSCrossRefGoogle Scholar
  46. [46]
    Villermaux, J.; Symposium on “Spatial Inhomogeneities and Transient Behaviour” Brussels, 1987.Google Scholar
  47. [47]
    Levenspiel, D.; “Chemical Reaction Engeneering”, Wiley, New York 1972.Google Scholar
  48. [48]
    Aris, R.; “Introduction to the Analysis of Chemical Reactors”, Prentice Hall, Englewood Cliffs, NJ, 1965.Google Scholar
  49. [49]
    Lapidus, L.; Amundson, N.R. Eds., “Chemical Reactor Theory. A Review”, Prentice Hall, Englewood Cliffs, NJ, 1977.Google Scholar
  50. [50]
    Pacault, A.; Hanusse, P.; DeKepper, P.; Vidal, C.; Boissonade, J.; Acc. Chem. Res., 1976, 9, 439.CrossRefGoogle Scholar
  51. [51]
    Graziani, K.R.; Hudson, J.L.; Schmitz, R.A.; Chem. Eng. J., 1976, 12, 9.CrossRefGoogle Scholar
  52. [52]
    Marek, M.; Stuchl, J.; Biophys. Chem., 1975, 3, 241.CrossRefGoogle Scholar
  53. [53]
    Schneider, F.W.; Biopolymers, 1976, 15, 1.CrossRefGoogle Scholar
  54. [54]
    Schneider, F.W.; Kraus, H. P.; to be published.Google Scholar
  55. [55]
    Ottino, J.M.; Leony, C.W.; Rising, H.; Swanson, P.D.; Nature, 1988, 333, 419.ADSCrossRefGoogle Scholar
  56. [56]
    Field, R.J.; Försterling, H.D.; J. Phys. Chem., 1986, 90, 5400.CrossRefGoogle Scholar
  57. [57]
    Roux, J.C., DeKepper, P., Boissonade, P., Phys. ett. A, 1983, 97A, 168.ADSGoogle Scholar
  58. [58]
    Ruoff, P.; Chem. Phys. Lett., 1982, 90, 76.ADSCrossRefGoogle Scholar
  59. [59]
    Menzinger, M., Jankowski, P., J. Phys. Chem., 1986, 90, 1217.CrossRefGoogle Scholar
  60. [60]
    Ruoff P., Noyes R.M., J. Phys. Chem., 1986, 90, 4700.CrossRefGoogle Scholar
  61. [61]
    Menzinger, M., Jankowski, P., J. Phys. Chem., 1986, 90, 6865.CrossRefGoogle Scholar
  62. [62]
    Luo, Y.; Epstein, I.R.; J. Chem. Phys., 1986, 85, 5733.ADSCrossRefGoogle Scholar
  63. [63]
    Maselko, J.; Swinney, H.L.; J. Chem. Phys., 1986, 85, 6430.ADSCrossRefMathSciNetGoogle Scholar
  64. [64]
    Procaccia, I.; Nature, 1988, 333, 618.ADSCrossRefGoogle Scholar
  65. [65]
    Bassett, M.R.; Hudson, J.L.; J. Chem. Phys., 1989, 93, 2731.CrossRefGoogle Scholar
  66. [66]
    Field, R.J.; Burger, M.; Eds. in “Oscillations and Travelling Waves in Chemical Systems” Wiley, New York 1985.Google Scholar
  67. [67]
    Wolf, A.; Swift, J.B.; Swinney, H.L.; Vastano, J.L.; Physica, 1985, 16D, 285.ADSMathSciNetGoogle Scholar
  68. [68]
    Kaplan, J.; Yorke, J.; in “Lecture Notes in Mathematics”, Vol. 730, p. 228, H.O. Pleitgen; H.O. Walter, Eds., Springer, Berlin 1978.Google Scholar
  69. [69]
    Kruel, Th.M.; Schneider, F.W.; submitted to J. Phys. Chem.Google Scholar
  70. [70]
    Freund, A.; Kruel, Th.M.; Schneider, F.W.; in “From Chemical to Biological Organization”, Markus, M.; Müller, S.C.; Nicolis. G.; (Eds.) Springer, Berlin 1988.Google Scholar
  71. [71]
    Schneider, F.W.; Kruel, Th.M.; Freund, A.; in “Structure, Coherence and Chaos in Dynamical Systems”, Christiansen, P.D.; Parmentier, R.D.; Eds., Manchester University Press, Manchester, New York, 1989.Google Scholar

Copyright information

© Springer-Verlag Berlin, Heidelberg 1990

Authors and Affiliations

  • F. W. Schneider
    • 1
  • A. F. Münster
    • 1
  1. 1.Institute of Physical ChemistryUniversity of WürzburgWürzburgFed. Rep. of Germany

Personalised recommendations