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New scenario of the transition to chaos in degenerated systems

Research Project

Project/Area Number 11837006
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research InstitutionUniversity of Fukui

Principal Investigator

MIKHAEL Tribelsky  Fukui University, Engineering, Professor, 工学部, 教授 (50311684)

Co-Investigator(Kenkyū-buntansha) OGAWA Atsushi  Engineering, Assistant, 工学部, 助手 (70242584)
Project Period (FY) 1999 – 2001
Project Status Completed (Fiscal Year 2001)
Budget Amount *help
¥3,000,000 (Direct Cost: ¥3,000,000)
Fiscal Year 2001: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 2000: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 1999: ¥1,200,000 (Direct Cost: ¥1,200,000)
Keywordspattern formation / symmetry / degeneracy / Goldstone modes / dynamical chaos / International Exchange of Information / Germany, Kazakhstan, Russia, Spain, USA / pattern formation / Goldstone modes / symmetry breaking bifurcation / chaos / turbulence / nonlinearity / パターン形成 / 短波長不安定性 / 縮退 / ゴールドスト-ンモード / カオス / リアプノフ次元 / エルゴード性
Research Abstract

During the term of the present grant a systematic study of the soft-mode turbulence (SMT) and related topics has been carried out. The following results have been obtained. In case of electroconvection in homeotropically aligned nematic layer SMT is caused by slow random long-wavelength modulations of a roll pattern. The temporal autocorrelation function for components of the order parameter is calculated and expressed in terms of probability density for random drift velocity of the pattern. It is shown that despite the problem has at least two different characteristic times associated with the slow pattern dynamics, only one of them enters into the autocorrelation function.
The simplest nonlinear equation exhibiting SMT is the so-called Nikolaevskii equation. We employ numerical integration of this model to obtain detailed quantitative description of SMT. It is shown that SMT is characterized by a smooth interplay of different spatial scales, with defect generation being unimportant. T … More he Lyapunov exponents are calculated for several system sizes for fixed values of the control parameter ε. The Lyapunov dimension and the Kolmogorov-Sinai entropy are calculated and both shown to exhibit extensive and microextensive scaling. The distribution functional is shown to satisfy Gaussian statistics at small wavenumbers and small frequency. It is shown that if such a system undergoes instability against spatially periodic perturbations with a finite wavenumber, interplay of short-wavelength modes associated with the instability and long-wavelengths modes generated by the symmetry transformation affects the dynamics of the system dramatically. In particular, it may result in direct transition from a spatially uniform state to SMT, analogous to second order phase transition in equilibrium systems. Deep connection between SMT and the structure of the the dispersion equation for the relevant stability problem is revealed. A general phenomenological theory of hydrodynamic waves in regions with smooth loss of convexity of isentropes is developed. The theory is based upon the fact that for most media these regions in the p-V plane are anomalously small. The corresponding generalized Burgers equation is derived and analyzed. The exact solution of the equation for steady shock waves of rare faction is obtained and discussed.
The dynamics of actual market prices is a very specific example of dynamical chaos. We apply our knowledge in the theory of dynamical chaos to quantitative analysis of this type of chaos. A number of particular examples (exchange rates USD vs. JPY, XAU vs. USD, oil, etc.) is considered in detail. It allows predicting the price dynamics in future with high accuracy. The main advantage of the approach is that the prediction error does not increase in the course of time. Less

Report

(4 results)
  • 2001 Annual Research Report   Final Research Report Summary
  • 2000 Annual Research Report
  • 1999 Annual Research Report
  • Research Products

    (15 results)

All Other

All Publications (15 results)

  • [Publications] M.I.Tribelsky: "Statistical properties of chaos at onset of electroconvection in a homeotropically aligned nematic layer"Phys. Rev. E. 59. 3729-3732 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H.Xi: "Extensivechaos in the Nikolaevskii model"Phys. Rev. E. (Rapid Comm.). 62. R17-R20 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] M.I.Tribelsky: "New type of turbulence, or how symmetry results in chaos"Macromol. Symp.. 160. 225-231 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] M.I.Tribelsky: "Hydrodynamic Waves in Regions with Smooth Loss of Convexity of Isentropes : General Phenomenological Theory"Phys. Rev. Lett.. 86. 4037-4040 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] M.I.Tribelsky: "Predictability of market prices"Empirical Science of Financial Fluctuations. The Advent of Econophysics, edited by H. Takayasu (Springer, Tokyo, Berlin, etc.). 241-249 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] M. I. Tribelsky: "Statistical properties of chaos at onset of electroconvection in a homeotropically aligned nematic layer"Pnys. Rev, E. 59. 3729-3732 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H. Xi, R. Toral, J. D. Gunton, and M. I. Tribelsky: "Extensive chaos in the Nikolaevskii model"Phys. Rev. E. (Rapid Comm.). 62. R17-R20 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] M. I. Tribelsky: "New type of turbulence, or how symmetry results in chaos"Macromol. Symp.. 160. 225-231 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] M. I. Tribelsky, and S. I. Anisimov: "Hydrodynamic Waves in Regions with Smooth Loss of Convexity of Isentropes: General Phenomenological Theory"Phys. Revj Lett.. 86. 4037-4040 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] M. Tribelsky, Y. Harada, N. Makarenko, and Y. Kuandykov: "Predictability of market prices"Empirical Science of Financial Fluctuations. The Advent of Econoohysics, edited by H. Takayasu (Springer, Tokyo, Berlin, etc.). 241-249 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] M.I.Tribelsky: "Hydrodynamic Waves in Regions with Smooth Loss of Convexity of Isentropes : General Phenomenological Theory"Physical Review Letters. 86. 4037-4040 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] M.Tribelsky: "Predictability of market prices"Empirical Science of Financial Fluctuations. The Advent of Econophysics, edited by H.Takayasu (Springer, Tokyo, Berlin, etc.,). 241-249 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] H.Xi: "Extensive Chaos in the Nikolaevskii Model"Phys.Rev.E.(Rapid Comm.). 62. 17-20 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] M.I.Tribelsky: "New Type of Turbulence, or How Symmetry Results in Chaos"Macromol.Symp.. 160. 225-232 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] M. I. Tribelsky: "New type of turbulence, or how symmetry results in chaos"Macromolecular Symposia. (発表予定、印刷準備中). (2000)

    • Related Report
      1999 Annual Research Report

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Published: 1999-04-01   Modified: 2016-04-21  

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