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New Development of Singular-Perturbation Method with Renormalization Group

Research Project

Project/Area Number 13640402
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field 物理学一般
Research InstitutionNagoya University

Principal Investigator

NOZAKI Kazuhiro  Nagoya University, Graduate School of Science, professor, 大学院・理学研究科, 教授 (00115619)

Co-Investigator(Kenkyū-buntansha) KONISI Teturo  Nagoya University, Graduate School of Science, associate professor, 大学院・理学研究科, 助教授 (30211238)
Project Period (FY) 2001 – 2004
Project Status Completed (Fiscal Year 2004)
Budget Amount *help
¥2,400,000 (Direct Cost: ¥2,400,000)
Fiscal Year 2004: ¥500,000 (Direct Cost: ¥500,000)
Fiscal Year 2003: ¥500,000 (Direct Cost: ¥500,000)
Fiscal Year 2002: ¥600,000 (Direct Cost: ¥600,000)
Fiscal Year 2001: ¥800,000 (Direct Cost: ¥800,000)
KeywordsSingular perturbation method / renormalization group / symplectic map / Lie symmetry / phase equation / similarity-type solution / Lie symmetry / 非線形偏微分方程式 / 不変解 / シンプレクティク写像 / ポアンカレーバーコフ分岐 / くり込み群の方法 / symplectic map / reduced map / 共鳴島形成 / Liouville expansion / 楕円点 / Lie symmery / functional self-similarity / 境界条件 / 自己重力中のガス球 / くりこみ群 / シンプレクティック・マップ / 反応拡散系
Research Abstract

The long-time asymptotic behavior of a nonlinear dynamical system has been studied by various singular perturbation methods, such as the multi-time method, the normal form theory and the reductive perturbation method. Those various perturbation methods have been unified by the recent development of the perturbative renormalization group method (the RG method), where the long-time asymptotic behavior of a system is described by the renormalization group equation. In this work, the RG method is developed as follows.
1.A symplecticity-preserving RG method is formulated and is applied to the Poicare-Birkhoff bifurcation of a two-dimensional symplectic map. We obtain analytical expressions to the resonant island structure.
2.A proto-RG approach of the RG method is proposed to avoid the step of obtaining explicit secular solutions in the RG method. Various phase equations are systematically derived as RG equations from a general reaction-diffusion equation.
3.The RG method with the Lie symmetry is extended to apply to interesting physical systems such as a gas sphere under gravity and adiavatic perfect gas, which have only trivial Lie symmetries.

Report

(5 results)
  • 2004 Annual Research Report   Final Research Report Summary
  • 2003 Annual Research Report
  • 2002 Annual Research Report
  • 2001 Annual Research Report
  • Research Products

    (37 results)

All 2004 2002 2001 Other

All Journal Article (25 results) Publications (12 results)

  • [Journal Article] Renormalization group Symmetry Method and Gas Dynamics2004

    • Author(s)
      S.Murata
    • Journal Title

      J.Non-linear Mech. 39

      Pages: 963-967

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Liouville Operator Approach to Symplecticity-Preserving Renormalization Group Method2004

    • Author(s)
      S.Goto
    • Journal Title

      Physica D 194

      Pages: 175-186

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Renormalization Analysis of Resonance Structure in a 2-D Symplectic Map2004

    • Author(s)
      T.Maruo
    • Journal Title

      Prog.Theor.Phys. 111(4)

      Pages: 463-474

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Dynamics of Gas Sphere under Self-Gravity.2004

    • Author(s)
      S.Murata
    • Journal Title

      Phys.Letters A332

      Pages: 35-38

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Renormalization group symmetry method and gas dynamics2004

    • Author(s)
      S.Murata, K.Nozaki
    • Journal Title

      J.Non-linear Mech. 39

      Pages: 963-967

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Liouville Operator Approach to Symplecticity-Preserving Renormal ization Group Method2004

    • Author(s)
      S.Goto, K.Nozaki
    • Journal Title

      Physica D 194

      Pages: 175-186

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Renormalization Analysis of Resonance Structure in a 2-D Symplectic Map2004

    • Author(s)
      T.Maruo, S.Goto, K.Nozaki
    • Journal Title

      Prog.Theor.Phys. 111(4)

      Pages: 463-474

    • NAID

      110001196158

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Dynamics of gas sphere under self-gravity2004

    • Author(s)
      S.Murata, K.Nozaki
    • Journal Title

      Phys.Letters A 332

      Pages: 35-38

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Liouville operator Approach to symplecticity-preserving RGM2004

    • Author(s)
      S.Goto, K.Nozaki
    • Journal Title

      Physica D 194

      Pages: 175-186

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Renormalization Analysis of Resonance Structure in a 2-D Symplectic Map2004

    • Author(s)
      T.Maruo, S.Goto, K.Nozaki
    • Journal Title

      Prog.Theor.Phys. 111・4

      Pages: 463-474

    • NAID

      110001196158

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Dynamics of gas sphere under self-gravity2004

    • Author(s)
      S.Murata, K.Nozaki
    • Journal Title

      Phys.Letters A332

      Pages: 35-38

    • Related Report
      2004 Annual Research Report
  • [Journal Article] A Phase Equation of Third-Order Spatial Derivative2002

    • Author(s)
      Y.Masutomi
    • Journal Title

      Prog.Theor.Phys. 107(2)

      Pages: 253-264

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Random Wandering around Homoelinic-like Manifolds in Symplectic Map Chain2002

    • Author(s)
      S.Goto
    • Journal Title

      Prog.Theor.Phys. 107(4)

      Pages: 637-654

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Exact Expanding Solution of Gas Sphere under Gravity.2002

    • Author(s)
      S.Murata
    • Journal Title

      J.Phys.Soc.Jpn 71(12)

      Pages: 2825-2826

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] A Phase Equation of Third-order Spatial Derivative2002

    • Author(s)
      Y.Masutomi, K.Nozaki
    • Journal Title

      Prog.Theor.Phys. 107(2)

      Pages: 253-264

    • NAID

      110001209311

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Random Wandering around Homoclinic-like manifolds in Symplectic Map Chain2002

    • Author(s)
      S.Goto, K.Nozaki, H.Yamada
    • Journal Title

      Prog.Theor.Phys. 107(4)

      Pages: 637-654

    • NAID

      130004540167

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Exact Expanding Solution of Gas Sphere under Gravity2002

    • Author(s)
      S.Murata, K.Nozaki
    • Journal Title

      J.Phys.Soc.Jpn 71(12)

      Pages: 2825-2826

    • NAID

      110001954145

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Regularized Renormalization Group Reduction of Symplectic Maps2001

    • Author(s)
      S.Goto
    • Journal Title

      J.Phys.Soc.Jpn 70(1)

      Pages: 49-54

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Asymptotic Expansions of Unstable and Stable Manifolds in Time-Discrete Systems2001

    • Author(s)
      S.Goto
    • Journal Title

      Prog.Theor.Phys. 105(1)

      Pages: 99-107

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Renormalization-Group Theoretical Reduction2001

    • Author(s)
      K.Nozaki
    • Journal Title

      Phys.Rev. E63(4)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Derivation of Non-isotropic Phase Equations from General Reaction-Diffusion Equation2001

    • Author(s)
      Y.Masutomi
    • Journal Title

      Physica D 151

      Pages: 44-60

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Regularized Renormalization Group Reduction of Symplectic Maps2001

    • Author(s)
      S.Goto, K.Nozaki
    • Journal Title

      J.Phys.Soc.Jpn. 70(1)

      Pages: 49-54

    • NAID

      110001971877

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Asymptotic Expansions of Unstable and Stable Manifolds in Time-Discrete Systems2001

    • Author(s)
      S.Goto, K.Nozaki
    • Journal Title

      Prog.Theor.Phys. 105(1)

      Pages: 99-107

    • NAID

      130004540043

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Renormalization-Group Theoretical Reduction2001

    • Author(s)
      K.Nozaki, Y.Oono
    • Journal Title

      Phys.Rev.E 63(4)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Derivation of Non-isotropic Phase Equations from General Reaction-Diffusion Equation2001

    • Author(s)
      Y.Masutomi, K.Nozaki
    • Journal Title

      Physica D 151

      Pages: 44-60

    • NAID

      110002190673

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Publications] Souichi Murata, K. Nozaki: "Renormalization Group Symmetry Method and Gas Dynamics"Int.J. Non_linear Mech.. 39. 963-967 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Souichi Murata: "RG Symmetry and Initial Value Problem for Shallow Water Equations"Chaos, Solitons and Fractals. 19. 809-812 (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] T.Yokoi, H.Yamada, K.Nozaki: "Exact Target and Spiral Pattern Solutions on Nonlinear Oscillations"J.Phys.Soc.Jpn.. (in press). (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] S.Goto, K.Nozaki: "Liouville Operator Approach to Symplecticity-Preserving RGM"Physica D. (in press). (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] T.Maruo, S.Goto, K.Nozaki: "Renormalization Analysis of Resonance Structure in 2-D S-Map"Prog.Theor.Phys.. (in press). (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] Sin-itiro Goto: "Random Wandering around Homoclinic-like manifolds in Symplectic Map Chain"Progress of Theretical Physics. 107・5. 637-674 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Yuji Masutomi: "A Phase Equation of Third-order Spatial Derivative"Progress of Theretical Physics. 107・2. 253-264 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Souichi Murata: "Exact Expanding Solution of Gas Sphere under Gravity"Journal of Physical Society of Japan. 71・12. 2825-2826 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Souichi Murata: "Renormalization Group Symmetry Method and Gas Dynamics"International Journal of Nonlinear Mechanics. (in press). (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] K.Nozaki: "Renormalization-group theoretical reduction"Physcal Review E. 63. 046101-1-046101-18 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Y.Masutomi: "A phase equation of third-order spetial derivetive"Progress of Theoretical Physics. 107・2(発表予定). (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] S.Goto: "Random wandering around howsclinic-like manifolds in symplectic map chain"Progress of Theoretical Physics. 107・4(発表予定). (2002)

    • Related Report
      2001 Annual Research Report

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Published: 2001-04-01   Modified: 2016-04-21  

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