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Bifurcation theoretical approach to chaotic dynamics and to systems with large degrees of freedom

Research Project

Project/Area Number 14340055
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Global analysis
Research InstitutionKYOTO UNIVERSITY

Principal Investigator

KOKUBU Hiroshi  Kyoto University, Department of Mathematics, Associate Professor, 大学院・理学研究科, 助教授 (50202057)

Co-Investigator(Kenkyū-buntansha) SHISHIKURA Mitsuhiro  Kyoto University, Department of Mathematics, Professor, 大学院・理学研究科, 教授 (70192606)
ASAOKA Masayuki  Kyoto University, Department of Mathematics, Lecturer, 大学院・理学研究科, 講師 (10314832)
ARAI Zin  Kyoto University, Department of Mathematics, Assistant Professor, 大学院・理学研究科, 助手 (80362432)
NISHIURA Yasumasa  Hokkaido University, Research Institute of Electronic Science, Professor, 電子科学研究所, 教授 (00131277)
TSUJII Masato  Hokkaido University, Department of Mathematics, Associate Professor, 大学院・理学研究科, 助教授 (20251598)
西田 孝明  京都大学, 大学院・理学研究科, 教授 (70026110)
津田 一郎  北海道大学, 大学院・理学研究科, 教授 (10207384)
岡本 久  京都大学, 数理解析研究所, 教授 (40143359)
深谷 賢治  京都大学, 大学院・理学研究科, 教授 (30165261)
柳田 英二  東北大学, 大学院・理学研究科, 教授 (80174548)
木坂 正史  京都大学, 大学院・人間・環境学研究科, 助教授 (70244671)
石井 豊  京都大学, 大学院・数理学研究院, 助手 (20304727)
Project Period (FY) 2002 – 2004
Project Status Completed (Fiscal Year 2004)
Budget Amount *help
¥14,100,000 (Direct Cost: ¥14,100,000)
Fiscal Year 2004: ¥3,900,000 (Direct Cost: ¥3,900,000)
Fiscal Year 2003: ¥5,100,000 (Direct Cost: ¥5,100,000)
Fiscal Year 2002: ¥5,100,000 (Direct Cost: ¥5,100,000)
Keywordsdynamical system / global structure / bifurcation / chaos / hyperbolicity / ergodic theory / complex dynamics / large degrees of freedom / 特異性 / 不変測度 / 大自由度系 / 特異構造 / 微分方程式 / カオス遍歴 / 摂動 / 複雑系 / 無限次元力学系 / 位相的方法
Research Abstract

Global structures and bifurcations of dynamical systems, with special emphasis on chaos-complicated and unpredictable behavior in dynamics-and systems of large degrees of freedom such as PDEs and coupled systems, are studied from various different points of view and many interesting results are obtained. As some of main results in this project, Kokubu (1)showed the existence of a singular invariant set called "singularly degenerate heteroclinic cycle" in the Lorenz system and its alike, from which a chaotic attractor of geometric Lorenz type is proven to bifurcate, (2)developed a theory describing the structure of singularly perturbed vector fields with using a topological invariant called Conley index, obtained a method to show the existence of periodic and chaotic solutions in such systems under suitable setting, and applied it to several concrete problems. Shishikura studied complex analytic dynamical systems, and in particular developed a renormalization theory for parabolic fixed points, which will be a new and very powerful tool for studying the structure and bifurcation of such systems. Asaoka studied dynamical systems with a sort of hyperbolicity called projectively Anosov structure and completed a classification in the case of 3-dimensional flows. Combining rigorous computation with topological methods such as the Conley index theory, Arai obtained several interesting results on hyperbolicity and global bifurcations in the Henon maps. Tsujii studied dynamical systems from ergodic theory viewpoint and obtained a general result on the existence of good invariant measures in 2-dimensional partially hyperbolic systems. Nishiura studied complicated interesting transient behavior observed in some kinds of PDEs called self-replicating and self-destruction patterns and clarified its mechanism by using dynamical system theory. Other results on systems with large degrees of freedom include Komuro's detailed analysis on chaotic itenerancy in globally coupled maps.

Report

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

    (26 results)

All 2005 2004 2003 2002 Other

All Journal Article (13 results) Book (2 results) Publications (11 results)

  • [Journal Article] A classification of three dimensional regular projectively Anosov flows2005

    • Author(s)
      M.Asaoka
    • Journal Title

      Proc.Japan Acad.Ser.A Math.Sci. 80

      Pages: 194-197

    • NAID

      40006712808

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Existence of a singularly degenerate heteroclinic cycle in the Lorenz system and its dynamical consequences, Part I2004

    • Author(s)
      Hiroshi Kokubu, Robert Roussarie
    • Journal Title

      Journal of Dynamics and Differential Equations 16

      Pages: 513-557

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Annual Research Report 2004 Final Research Report Summary
  • [Journal Article] Linearity of exceptional set for maps of P_k(C)2004

    • Author(s)
      J.-Y.Briend, S.Cantat, M.Shishikura
    • Journal Title

      Math.Annalen 330

      Pages: 39-43

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Linearity of the exceptional set for maps of P_k(C)2004

    • Author(s)
      J.-Y.Briend, S.Cantat, M.Shishikura
    • Journal Title

      Math.Annalen 330

      Pages: 39-43

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Global in time behavior of viscous surface waves : horizontally periodic motion2004

    • Author(s)
      Takaaki Nishida, Yoshiaki Teramoto, Hideaki Yoshihara
    • Journal Title

      Journal of Mathematics of Kyoto University 44-2

      Pages: 271-323

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Linearity of the exceptional set for maps of P_k(C)2004

    • Author(s)
      J.-Y.Briend, S.Cantant, M.Shishikura
    • Journal Title

      Math.Annalen 330

      Pages: 39-43

    • Related Report
      2004 Annual Research Report
  • [Journal Article] A classification of three dimensional regular projectively Anosov flows2004

    • Author(s)
      Masayuki Asaoka
    • Journal Title

      Proc.Japan Acad.Ser.A Math.Sci. 80-10

      Pages: 194-197

    • NAID

      40006712808

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Equivalence of graded module braids and interlocking sequences2003

    • Author(s)
      Zin Arai
    • Journal Title

      Journal of Mathematics of Kyoto University 43

      Pages: 441-449

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Chaotic solutions in slowly varying perturbations of Hamiltonian systems with applications to shallow water sloshing2002

    • Author(s)
      T.Gedeon, H.Kokubu, K.Mischaikow, H.Oka
    • Journal Title

      Journal of Dynamics and Differential Equations 14

      Pages: 63-84

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Physical measures for partially hyperbolic surface endomorphisms

    • Author(s)
      Masato TSUJII
    • Journal Title

      Acta mathematica (発行予定)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Physical measures for partially hyperbolic surface endomorphisms

    • Author(s)
      Masato TSUJII
    • Journal Title

      Acta mathematics (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Physical measures for partially hyperbolic surface endomorphisms

    • Author(s)
      Masato Tsujii
    • Journal Title

      Acta mathematica (発表予定)

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Smoothness of solenoidal attractors

    • Author(s)
      Artur Avila, Sebastien Gouezel, Masato Tsujii
    • Journal Title

      Discrete and Continuous Dynamical Systems (発表予定)

    • Related Report
      2004 Annual Research Report
  • [Book] 自己複製と自己崩壊のパターンダイナミクス,岩波講座「物理の世界」2003

    • Author(s)
      西浦廉政
    • Total Pages
      84
    • Publisher
      岩波書店
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Book] Pattern dynamics of self-replication and self-destruction2003

    • Author(s)
      Yasumasa Nishiura
    • Total Pages
      84
    • Publisher
      Iwanami
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Publications] Fukaya, Kenji: "Galois symmetry on Floer cohomology"Turkish Journal of Mathematics. 27. 11-32 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] S.Kim, H.Okamoto: "Bifurcations and inviscid limit of rhombic Navier-Stokes flows in tori"IMA Journal of Applied Mathematics. 68. 119-134 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] T.Umemura, I.Tsuda: "Chaotic itinerancy generated by coupling of Milnor attractors"Chaos. 13. 937-946 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] J.Buzzi, O.Sester, M.Tsujii: "Weakly expanding skew-products of quadratic maps"Ergodic Theory and Dynamical Systems. 23. 1401-1414 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] M.Kuwamura, E.Yanagida: "The Eclhaus and zigzag instability criteria in gradient/skew gradient dissipative systems"Physica D. 175. 185-195 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] T.Gedeon, H.Kokubu, K.Mischaikow, H.Oka: "Chaotic solutions in slowly varying perturbations of Hamiltonian systems with applications to shallow water sloshing"Journal of Dynamics and Differential Equations. 14. 63-84 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] S.Matsumoto, M.Shishikura: "Minimal sets of certain annular homeomorphisms"Hiroshima Mathematical Journal. 32. 207-215 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] M.Tsujii: "Absolutely continuous invariant measures for multidimensional piecewise expanding maps"Sugaku. 54. 383-397 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] A.Shudo, Y.Ishii, K.S Ikeda: "Julia set describes quantum tunnelling in the presence of chaos"J.Phys.A : Math.Gen.. 35. L225-L231 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] M.Shishikura: "Rigidity of quadratic polynomials"数理解析研究所講究録. 1269. 63-73 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] E.Yanagida: "Mini-maximizers for reaction-diffusion systems with skew-gradient structures"Journal of Differential Equations. 179. 311-335 (2002)

    • Related Report
      2002 Annual Research Report

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Published: 2002-04-01   Modified: 2016-04-21  

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