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Prog. Theor. Phys. Vol. 81 No. 1 (1989) pp. 60-77

[ Full Text PDF : FREE ACCESS (847K) ]

Scaling Structures and Statistical Mechanics of Type I Intermittent Chaos

Nobuyuki Mori, Tatsuharu Kobayashi, Hiroki Hata, Terumitsu Morita, Takehiko Horita and Hazime Mori

Department of Physics, Kyushu University 33, Fukuoka 812

(Received July 21, 1988)

Abstract:

Intermittent chaos exhibits regular laminar motions and irregular turbulent bursts alternately, indicating that its chaotic attractor has two different types of local structures. For type I intermittency just before the saddle-node bifurcation, it is shown that the two types of local structures can be captured by the fluctuation spectrum h(Λ) of the coarse-grained local expansion rates Λ of nearby orbits and their q-weighted average Λ(q), (-∞< q < ∞). The spectrum h(Λ) and the average Λ(q) are obtained analytically for a piecewise linear Markov map which exhibits type I intermittency, and numerically for the logistic map just before the period-three window. Thus it turns out that h(Λ) has a linear part between Λ= Λ1 > 0 and Λ= Λ2= 0 with a slope qδ, leading to a discontinuous transition of Λ(q) from Λ1 to Λ2 at q = qδ as q is increased across qδ. This represents a phase transition between the laminar motions and the turbulent bursts, and gives a new type of q-phase transition with 1.0 > qδ > 0.5 in contrast to other three types of q-phase transitions with transition points qα = 2.0, qβ < 0.5 and qγ = 1.0. Thus statistical mechanics of chaotic attractors at the bifurcation points is fully developed for the nontrivial case.


URL : http://ptp.ipap.jp/link?PTP/81/60/
DOI : 10.1143/PTP.81.60

[ Full Text PDF : FREE ACCESS (847K) ] Citation:

References:

  1. P. Bergé, Y. Pomeau and C. Vidal, Order within Chaos (Wiley, New York, 1984).
    H. G. Schuster, Deterministic Chaos (VCH, Weinheim, 1988).
  2. R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lect. Notes in Math. 470 (Springer, New York, 1975).
    O. Lanford, Entropy and Equilibrium States in Classical and Statistical Mechanics, in; Statistical Mechanics and Mathematical Problems, ed. A. Lenard, Lect. Notes in Phys. 20 (Springer, New York, 1976).
    D. Ruelle, Thermodynamic Formalism (Addison-Wesley, 1978).
  3. J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields (Springer-Verlag, 1983).
  4. T. C. Halsey, M. H. Jensen, L. P. Kadanoff, I. Procaccia and B. I. Shraiman, Phys. Rev. A33 (1986), 1141[APS].
    M. H. Jensen, L. P. Kadanoff, A. Libchaber, I. Procaccia and J. Stavans, Phys. Rev. Lett. 55 (1985), 2798[APS].
  5. M. Sano, S. Sato and Y. Sawada, Prog. Theor. Phys. 76 (1986), 945[PTP].
  6. J.-P. Eckmann and I. Procaccia, Phys. Rev. A34 (1986), 659[APS].
    H. Fujisaka and M. Inoue, Prog. Theor. Phys. 77 (1987), 1334[PTP].
    T. Bohr and D. Rand, Physica 25D (1987), 387[Elsevier].
    G. Paladin and A. Vulpiani, Phys. Rep. 156 (1987), 147[CrossRef].
  7. T. Morita, H. Hata, H. Mori, T. Horita and K. Tomita, Prog. Theor. Phys. 78 (1987), 511[PTP]; ibid. 79 (1988), 296[PTP].
  8. H. Hata, T. Horita, H. Mori, T. Morita and K. Tomita, Prog. Theor. Phys. 80 (1988), 809[PTP].
    T. Horita, H. Hata, H. Mori, T. Morita, K. Tomita, S. Kuroki and H. Okamoto, Prog. Theor. Phys. 80 (1988), 793[PTP].
    K. Tomita, H. Hata, T. Horita, H. Mori and T. Morita, Prog. Theor. Phys. 80 (1988), 953[PTP].
  9. T. Horita, H. Hata, H. Mori, T. Morita and K. Tomita, Prog. Theor. Phys. 81 (1989), 1[PTP].
  10. Y. Pomeau and P. Manneville, Commun. Math. Phys. 74 (1980), 189[CrossRef].
  11. J. E. Hirsch, B. A. Huberman and D. J. Scalapino, Phys. Rev. A25 (1982), 519[APS].
  12. H. Mori, H. Okamoto, B. C. So and S. Kuroki, Prog. Theor. Phys. 76 (1986), 784[PTP].
  13. N. Mori, S. Kuroki and H. Mori, Prog. Theor. Phys. 79 (1988), 1260[PTP].
  14. K. Shobu, T. Ose and H. Mori, Prog. Theor. Phys. 71 (1984), 458[PTP].
  15. H. Fujisaka and M. Inoue, Prog. Theor. Phys. 78 (1987), 268[PTP].
  16. H. Fujisaka, Prog. Theor. Phys. 70 (1983), 1264[PTP]; ibid. 71 (1984), 513[PTP].
  17. R. S. Ellis, Entropy, Large Deviations, and Statistical Mechanics (Springer-Verlag, New York, 1985).
  18. M. Dubois, M. A. Rubio and P. Bergé, Phys. Rev. Lett. 51 (1983), 1446[APS].
  19. H. Okamoto, H. Mori and S. Kuroki, Prog. Theor. Phys. 79 (1988), 581[PTP].
  20. P. Szépfalusy, T. Tél, A. Csordás and Z. Kovács, Phys. Rev. A36 (1987), 3525[APS].

Citing Article(s) :

  1. Progress of Theoretical Physics Vol. 81 No. 1 (1989) pp. 11-16 :
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    Hiroki Hata, Takehiko Horita, Hazime Mori, Terumitsu Morita and Koji Tomita
  2. Progress of Theoretical Physics Vol. 81 No. 6 (1989) pp. 1124-1134 :
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  3. Progress of Theoretical Physics Vol. 82 No. 1 (1989) pp. 1-6 :
    Critical Scaling Laws of Dynamic Structure Functions for Type I Intermittent Chaos
    Tatsuharu Kobayashi, Nobuyuki Mori, Hiroki Hata, Tekehiko Horita, Takeshi Yoshida and Hazime Mori
  4. Progress of Theoretical Physics Vol. 82 No. 5 (1989) pp. 863-868 :
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  5. Progress of Theoretical Physics Vol. 82 No. 5 (1989) pp. 879-896 :
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  6. Progress of Theoretical Physics Vol. 82 No. 5 (1989) pp. 897-910 :
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    Hiroki Hata, Takehiko Horita and Hazime Mori
  7. Progress of Theoretical Physics Vol. 83 No. 4 (1990) pp. 649-654 :
    q-Phase Transitions and Dynamic Scaling Laws at Attractor-Merging Crises in the Driven Damped Pendulum
    Toru Murayama, Hirotaka Tominaga, Hazime Mori, Hiroki Hata and Takehiko Horita
  8. Progress of Theoretical Physics Vol. 84 No. 4 (1990) pp. 553-557 :
    Dynamic Scaling Laws for Bifurcations of Chaos in Dissipative Differential Systems
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  9. Progress of Theoretical Physics Vol. 85 No. 2 (1991) pp. 187-204 :
    New Asymptotic Law Governing Overall Temporal Correlations in Chaotic Systems
    Hirokazu Fujisaka and Hiroshi Shibata
  10. Progress of Theoretical Physics Supplement No.99 (1989) pp. 1-63 :
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  11. Progress of Theoretical Physics Supplement No.99 (1989) pp. 64-81 :
    Global Bifurcations and Fluctuation Spectra of Local Expansion Rates in Nonlinear Dynamical Systems
    Takeshi Yoshida and Syuji Miyazaki
  12. Progress of Theoretical Physics Supplement No.99 (1989) pp. 82-94 :
    “Thermodynamics”, Temporal Correlations and Scaling Laws
    Hirokazu Fujisaka and Masayoshi Inoue

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