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Prog. Theor. Phys. Vol. 76 No. 2 (1986) pp. 335-355

[ Full Text PDF : FREE ACCESS (977K) ]

Phase Description Method to Time Averages in the Lorenz System

Shinji Koga

Department of Physics, Osaka Kyoiku University, Osaka 543

(Received March 5, 1986)

Abstract:

On the basis of the transformation to the rotating coordinates associated with the imbedded unstable limit cycles in the Lorenz system, we present a new representation for the long time averages, which may be applicable to any three-dimensional dissipative dynamical systems producing chaos. By employing the dynamics of the phases of the imbeddid limit cycles, we show that the time average is expressible in terms of two types of the weight factors; the residence time probability density and the factor inbersely proportional to the speed of the phase.


URL : http://ptp.ipap.jp/link?PTP/76/335/
DOI : 10.1143/PTP.76.335

[ Full Text PDF : FREE ACCESS (977K) ] Citation:

References:

  1. E. N. Lorenz, J. Atomos. Sci. 20 (1963), 130.
  2. B. Saltzman, J. Atomos. Sci. 19 (1962), 329.
  3. J. A. Yorke and E. D. Yorke, J. Stat. Phys. 21 (1979), 263.
  4. Y. Pomeau and P. Manneville, Commun. Math. Phys. 74 (1980), 189[CrossRef].
  5. K. Tomita and I. Tsuda, Prog. Theor. Phys. Suppl. No. 69 (1980), 185[PTP].
  6. I. Shimada and T. Nagashima, Prog. Theor. Phys. 59 (1978), 1033[PTP].
  7. M. Lücke, J. Stat. Phys. 15 (1976), 455.
  8. E. Knobloch, J. Stat. Phys. 20 (1979), 695.
  9. J. Guckenheimer, The Hopf Bifurcation and Its Application, ed. J. E. Marsden and M. J. McCracken (Springer, Berlin, 1976), p. 368.
  10. G. Ioos and D. D. Josepf, Elementary Stability and Bifurcation Theory (Springer, Berlin, 1980).
  11. T. Kai and K. Tomita, Prog. Theor. Phys. 64 (1980), 1532[PTP].
    Kai and Tomita have pointed out the important role of the imbedded unstable cycles in determining the structures of the chaos in one-dimensional difference systems. Especially, the cycle with considerably large period is emphasized in their considerations.
  12. Some of the limit cycles of the same topological property become stable for the large value of the parameter r. For example, T. Shimazu, Physica 97A (1979), 383, and Ref. 5).

Citing Article(s) :

  1. Progress of Theoretical Physics Vol. 77 No. 5 (1987) pp. 1057-1076 :
    Phase Description Method to Time Average in Dissipative Chaos
    Shinji Koga
  2. Progress of Theoretical Physics Vol. 80 No. 2 (1988) pp. 216-231 :
    Phase Description Method to Invariant Densities in 1-D Difference Systems
    Shinji Koga

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