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Prog. Theor. Phys. Vol. 63 No. 1 (1980) pp. 91-105

[ Full Text PDF : FREE ACCESS (1139K) ]

Dynamics of the Lorenz Model of Convective Instabilities. II

Kyôzô Takeyama

Department of Physics, Chuo University, Tokyo 112

(Received July 13, 1979)

Abstract:

The Lorenz model of convective instabilities is reduced to a one-dimensional dynamics of a fictitious particle with memory (and friction) in a potential well. When the potential has double minima, memory-controlled oscillations can lead to apparently erratic behaviors. In some simplifying situations, the relationship in the magnitude of two successive amplitudes is shown to exhibit a cusp singularity which is observed in numerical experiments.


URL : http://ptp.ipap.jp/link?PTP/63/91/
DOI : 10.1143/PTP.63.91

[ Full Text PDF : FREE ACCESS (1139K) ] Citation:

References:

  1. C. Normand, Y. Pomeau and M. G. Velarde, Rev. Mod. Phys. 49 (1977), 581[APS].
  2. F. H. Busse, Rep. Prog. Phys. 41 (1978), 1929[CrossRef].
  3. B. Saltzman, J. Atmos. Sci. 19 (1962), 329.
  4. E. N. Lorenz, J. Atmos. Sci. 20 (1963), 130.
  5. P. C. Martin, J. de Phys. 37 (1976), Colloq. Cl. 57, or in Statistical Physics, edited by L. Pál and P. Szépfalusy (North-Holland, Amsterdam, 1976), p. 69.
  6. D. D. Joseph, Stability of Fluid Motions (Springer, Berlin, 1976), vol. I, p. 60.
  7. H. L. Swinney and J. P. Gollub, Phys. Today 31 (1978), 41.
  8. J. B. McLaughlin and P. C. Martin, Phys. Rev. A 12 (1975), 186[APS].
  9. H. Haken, Phys. Lett. A 53 (1975), 77[CrossRef].
    R. Graham, Phys. Lett. A 58 (1976), 440[CrossRef].
  10. K. A. Robbins, Math. Proc. Camb. Phil. Soc. 82 (1977), 309.
  11. P. C. Martin, Physica A 96 (1979), 70 [CrossRef](see especially Fig. 7).
  12. M. Lücke, J. Stat. Phys. 15 (1976), 455.
    E. Knobloch, J. Stat. Phys. 20 (1979), 695.
  13. M. Hénon and Y. Pomeau, in Turbulence and Navier-Stokes Equation, Orsay 1975: Lecture Notes in Mathematics 565, edited by A. Dold and B. Eckmann (Springer, Berlin, 1976), p. 29.
    D. Ruelle, in Turbulence and Navier-Stokes Equation, Orsay 1975: Lecture Notes in Mathematics 565, edited by A. Dold and B. Eckmann (Springer, Berlin, 1976), p. 146.
    M. Hénon, Commun. Math. Phys. 50 (1976), 69[CrossRef].
    S. D. Feit, Commun. Math. Phys. 61 (1978), 249[CrossRef].
    Y. Pomeau, in Problems of Stellar Convection, Nice 1976: Lecture Notes in Physics 71, edited by J. Ehlers et al. (Springer, Berlin, 1977), p. 337.
    J. L. Ibañez and Y. Pomeau, J. Non-Equilib. Thermodyn. 3 (1978), 135.
  14. J. E. Marsden and M. McCracken, The Hopf Bifurcation and Its Applications (Springer, Berlin, 1976), p. 141.
    J. Guckenheimer, in The Hopf Bifurcation and Its Applications by J. E. Marsden and M. McCracken, (Springer, Berlin, 1976), p. 368.
  15. D. Ruelle, in Statistical Mechanics and Dynamical Systems....: Duke University Mathematics Series III, edited by W. K. Allard et al. (Duke University, Durham, 1976), p. 1.
    O. E. Lanford III, in Statistical Mechanics and Dynamical Systems....: Duke University Mathematics Series III, edited by W. K. Allard et al. (Duke University, Durham, 1976), p. IV. 1.
  16. J. E. Marsden, in Turbulence Seminar, Berkeley 1976/77: Lecture Notes in Mathematics 615, edited by A. Dold and B. Eckmann (Springer, Berlin, 1977), p. 1.
    R. F. Williams, in Turbulence Seminar Berkeley 1976/77: Lecture Notes in Mathematics 615, edited by A. Dold and B. Eckmann (Springer, Berlin, 1977), p. 94.
  17. O. E. Rössler, Phys. Lett. A 57 (1976), 397[CrossRef]; ibid. 60 (1977), 392[CrossRef]; in Synergetics, edited by H. Haken (Springer, Berlin, 1977), p. 184.
    P. Coullet, C. Tresser and A. Arnéodo, Phys. Lett. A 72 (1979), 268[CrossRef].
  18. H. Haken and A. Wunderlin, Phys. Lett. A 62 (1977), 133[CrossRef].
  19. Y. Aizawa and I. Shimada, Prog. Theor. Phys. 57 (1977), 2146[PTP].
    T. Nagashima and I. Shimada, Prog. Theor. Phys. 58 (1977), 1318[PTP].
    I. Shimada and T. Nagashima, Prog. Theor. Phys. 59 (1978), 1033[PTP]; ibid. 61 (1979), 1605[PTP].
    I. Shimada, Prog. Theor. Phys. 62 (1979), 61[PTP].
  20. T. Shimizu and N. Morioka, Phys. Lett. A 66 (1978), 182[CrossRef]; ibid. 69 (1978), 148[CrossRef].
    N. Morioka and T. Shimizu, Phys. Lett. A 66 (1978), 447[CrossRef].
    T. Shimizu, Physica A 97 (1979), 383[CrossRef].
  21. T. Yamada and T. Nakano, unpublished (Chuo University, 1978).
  22. K. Takeyama, Prog. Theor. Phys. 60 (1978), 613[PTP].
  23. J. L. Kaplan and J. A. Yorke (p. 400), O. E. Rössler (p. 376) and R. F. Williams (p. 393), in Bifurcation Theory and Applications in Scientific Disciplines (Ann. The New York Academy of Sciences, vol. 316), edited by O. Gurel and O. E. Rössler (The New York Academy of Sciences, New York, 1979).
  24. C. J. Marzec and E. A. Spiegel, Preprint (Columbia University).
  25. F. Čap and H. Lashinsky, Nonlinear Vib. Prob. (Zagadnienia Drgań Nieliniowich) 14 (1973), 519.
  26. G. Oster and J. Guckenheimer, in The Hopf Bifurcation and Its Applications, by J. E. Marsden and M. McCracken (Springer, Berlin, 1976), p. 327.
    R. M. May and G. F. Oster, American Naturalist 110 (1976), 573.
    Y. Oono, Prog. Theor. Phys. 59 (1978), 1029[PTP].
  27. P. J. Holmes, J. Sound and Vibration 53 (1977), 471 (see especially Fig. 14).
  28. P. J. Holmes and D. A. Rand, J. Sound and Vibration 44 (1976), 237.
    Y. Ueda, J. Stat. Phys. 20 (1979), 181[CrossRef].
  29. A. E. Cook and P. H. Roberts, Proc. Camb. Phil. Soc. 68 (1970), 547.
  30. K. Takeyama, to be published.

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