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カオス力学系における不安定周期解の統計的性質

Research Project

Project/Area Number 19654019
Research Category

Grant-in-Aid for Challenging Exploratory Research

Allocation TypeSingle-year Grants
Research Field General mathematics (including Probability theory/Statistical mathematics)
Research InstitutionKyoto University

Principal Investigator

山田 道夫  Kyoto University, 数理解析研究所, 教授 (90166736)

Co-Investigator(Kenkyū-buntansha) 竹広 真一  京都大学, 数理解析研究所, 准教授 (30274426)
Project Period (FY) 2007 – 2009
Project Status Completed (Fiscal Year 2009)
Budget Amount *help
¥3,300,000 (Direct Cost: ¥3,300,000)
Fiscal Year 2009: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2008: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2007: ¥1,400,000 (Direct Cost: ¥1,400,000)
Keywords乱流シェルモデル / 不安定周期軌道 / 軌道平均値 / リヤプノフ解析 / ベータ平面
Research Abstract

本年度は前年度に引き続いて,カオス解をもつ力学系の不安定周期軌道を数値的に検出し,得られた周期軌道に沿った物理量の軌道平均値を求め統計的に検討した.従来の研究から,同程度の長さの周期をもつ不安定周期軌道による軌道平均値の分散は,周期依存性が非常に小さいことを見出していたが,小さな系においてこの結果を確認する数値計算を行うとともに,Kuramoto-Sivashinsky方程式のカオス解に対しても実行し,同じ性質を示すことを見出した.また乱流シェルモデルのカオス解に対して,最近Ginelliらが提案した共変リヤプノkフ解析を実行した.この解析は解軌道上の点における安定多様体と不安定多様体の方向を決定し系の双曲/非双曲性の程度の判別を可能にするため,比較的低次元で物理的性質を反映する乱流のシェルモデルに適用して,双曲/非双曲性と物理的性質の関連を調べた.その結果,解の間欠性にともなうバーストの発生が安定多様体/不安定多様体が接触(あるいはほとんど接触)する点の通過後に発生することを示唆する結果が得られた.この結果も含め,リヤプノフベクトルの構造と乱流の物理的構造の関連,また偏微分方程式系など大自由度カオスにおける双曲/非双曲性の構造は本研究計画後の研究課題である.

Report

(3 results)
  • 2009 Annual Research Report
  • 2008 Annual Research Report
  • 2007 Annual Research Report
  • Research Products

    (26 results)

All 2010 2009 2008 2007

All Journal Article (7 results) (of which Peer Reviewed: 7 results) Presentation (19 results)

  • [Journal Article] Closed vortex in a rotating polar cap2010

    • Author(s)
      Y.Taniguchi
    • Journal Title

      Theoretical and Applied Mechanics Japan 58

      Pages: 131-143

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Time averaged properties along unstable periodic orbits and chaotic orbits in ordinary differential equation systems2009

    • Author(s)
      Y.Saiki
    • Journal Title

      Physical Review E 79

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Stability of flow on a rotating polar cap2009

    • Author(s)
      Y.Taniguchi
    • Journal Title

      Fluid Dynamics Research 41

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Time averaged properties along unstable periodic orbits And chaot ic orbits in ordinary differential equation systems2009

    • Author(s)
      Y. Saiki, M. Yamada
    • Journal Title

      Physical Review E 79

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Time averaged properties along unstable periodic orbits and chaot ic orbits in two map systems.2008

    • Author(s)
      Y. Saiki, M. Yamada
    • Journal Title

      Nonlinear Process in Geophysics 15

      Pages: 675-680

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Chaotic properties of a fully developed turbulence2007

    • Author(s)
      M. Yamada
    • Journal Title

      Nonlinear Pocesses in Geophysics 14-5

      Pages: 631-640

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Rossby waves and jets in a two-dimensional decaying turbulence on a rotating sphere.2007

    • Author(s)
      Y. -Y. Hayashi
    • Journal Title

      Journal of the Atomospheric Sciences 64-12

      Pages: 4246-4269

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Presentation] 2次元 Navier-Stokes 方程式における乱流化について2010

    • Author(s)
      山田道夫
    • Organizer
      北陸応用数理研究会2010
    • Place of Presentation
      金沢大学
    • Year and Date
      2010-02-13
    • Related Report
      2009 Annual Research Report
  • [Presentation] Mechanism of a temporal intermittency in the GOY shell model turbulence2010

    • Author(s)
      Miki U.Kobayashi, M.Yamada
    • Organizer
      Exploring Complex Dynamics in High-Dimensional Chaotic Systems
    • Place of Presentation
      Max Planck Institute, Germany
    • Year and Date
      2010-01-15
    • Related Report
      2009 Annual Research Report
  • [Presentation] シェルモデル乱流における間欠性のメカニズムについて2009

    • Author(s)
      小林幹, 山田道夫
    • Organizer
      「非線形の科学」第4回
    • Place of Presentation
      京都大学
    • Year and Date
      2009-09-29
    • Related Report
      2009 Annual Research Report
  • [Presentation] シェルモデル乱流における間欠性と非双曲性22009

    • Author(s)
      小林幹, 山田道夫
    • Organizer
      日本物理学会第64回秋季大会
    • Place of Presentation
      熊本大学
    • Year and Date
      2009-09-28
    • Related Report
      2009 Annual Research Report
  • [Presentation] Time averaged properties along unstable periodic orbits in some systems of differential equations2009

    • Author(s)
      Y.Saiki, M.Yamada
    • Organizer
      The Third International Symposium on Recurrence Plots
    • Place of Presentation
      University of Montreal, Canada
    • Year and Date
      2009-08-26
    • Related Report
      2009 Annual Research Report
  • [Presentation] 乱流の少数自由度モデルと力学系的記述2009

    • Author(s)
      山田道夫
    • Organizer
      数理解析研究所研究集会「双曲力学系から大自由度力学系へ」
    • Place of Presentation
      京都大学
    • Year and Date
      2009-08-20
    • Related Report
      2009 Annual Research Report
  • [Presentation] 大自由度力学系としての乱流2009

    • Author(s)
      山田道夫
    • Organizer
      数理解析研究所研究集会「双曲力学系から大自由度力学系へ」
    • Place of Presentation
      京都大学
    • Year and Date
      2009-08-18
    • Related Report
      2009 Annual Research Report
  • [Presentation] Two-dimensional turbulence on a rotating sphere and the beta-plane2009

    • Author(s)
      山田道夫
    • Organizer
      Lecture Meeting : Hasegawa-Mima equation, 30 years and future
    • Place of Presentation
      京都大学
    • Year and Date
      2009-05-21
    • Related Report
      2009 Annual Research Report
  • [Presentation] Two-dimensional turbulence on a rotating sphere2009

    • Author(s)
      M. Yamada
    • Organizer
      The 5^<th> GN workshop : Workshop on Wave Turbulence
    • Place of Presentation
      Gifu University
    • Year and Date
      2009-03-27
    • Related Report
      2008 Annual Research Report
  • [Presentation] Time averaged properties along unstable periodic orbits in some systems of ordinary differential equations2009

    • Author(s)
      Y. Saiki, M. Yamada
    • Organizer
      第5回数学総合若手研究集会
    • Place of Presentation
      北海道大学
    • Year and Date
      2009-03-06
    • Related Report
      2008 Annual Research Report
  • [Presentation] 回転球面上の2次元流体運動について2009

    • Author(s)
      山田道夫
    • Organizer
      北陸応用数理研究会2009(拡大HMCセミナー)
    • Place of Presentation
      金沢大学サテライトプラザ
    • Year and Date
      2009-02-12
    • Related Report
      2008 Annual Research Report
  • [Presentation] Time averaged properties along unstable periodic orbits in some systems of differential equations2009

    • Author(s)
      Y. Saiki, M. Yamada
    • Organizer
      Sapporo Winter School
    • Place of Presentation
      北海道大学
    • Year and Date
      2009-02-12
    • Related Report
      2008 Annual Research Report
  • [Presentation] Time averaged properties along unstable periodic orbits and chaotic orbits in some systems of differential equations2009

    • Author(s)
      Y. Saiki, M. Yamada
    • Organizer
      Applied dynamics seminar
    • Place of Presentation
      Maryland University (USA)
    • Year and Date
      2009-01-15
    • Related Report
      2008 Annual Research Report
  • [Presentation] Time Averaged Properties along Unstable Periodic Orbits in Some Systems of Ordinary Differential Equations2009

    • Author(s)
      Y. Saiki, M. Yamada
    • Organizer
      Dynamics Days 2009
    • Place of Presentation
      San Diego (USA)
    • Year and Date
      2009-01-10
    • Related Report
      2008 Annual Research Report
  • [Presentation] 低次元常微分方程式系における軌道平均値について2008

    • Author(s)
      Y. Saiki, M. Yamada
    • Organizer
      京都力学系セミナー
    • Place of Presentation
      京都大学
    • Year and Date
      2008-12-05
    • Related Report
      2008 Annual Research Report
  • [Presentation] カオス力学系の不安定周期軌道解析-統計量と数値計算-2008

    • Author(s)
      Y. Saiki, M. Yamada
    • Organizer
      東京大学工学部航空宇宙工学非線形セミナー
    • Place of Presentation
      東京大学
    • Year and Date
      2008-11-25
    • Related Report
      2008 Annual Research Report
  • [Presentation] 第二世代ウェーブレット2008

    • Author(s)
      山田道夫
    • Organizer
      ウェーブレットの構成法と理工学的応用
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2008-10-28
    • Related Report
      2008 Annual Research Report
  • [Presentation] カオス力学系の不安定周期軌道解析に関する大規模数値解析2008

    • Author(s)
      Y. Saiki, M. Yamada
    • Organizer
      東工大数学専攻セミナー
    • Place of Presentation
      東京工業大学
    • Year and Date
      2008-07-24
    • Related Report
      2008 Annual Research Report
  • [Presentation] 2D Navier-Stokes flow on a rotating sphere2008

    • Author(s)
      M. Yamada
    • Organizer
      京都大学数理解析研究所合宿セミナー「数理流体力学:抽象論と計算力学的手法の融合」
    • Place of Presentation
      Kobe Institute
    • Year and Date
      2008-03-18
    • Related Report
      2007 Annual Research Report

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Published: 2007-04-01   Modified: 2016-04-21  

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