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Topological and computational methods for chaos and global structure of dynamical systems

Research Project

Project/Area Number 14540219
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Global analysis
Research InstitutionRyukoku University

Principal Investigator

OKA Hiroe  Ryukoku University, Faculty of Sciences and Technology, Professor, 理工学部, 教授 (20215221)

Co-Investigator(Kenkyū-buntansha) ITO Toshikazu  Ryukoku University, Faculty of Economics, Professor, 経済学部, 教授 (60110178)
YAMAGISHI Yoshikazu  Ryukoku University, Faculty of Sciences and Technology, Assistant Professor, 理工学部, 助手 (40247820)
MATSUOKA Takashi  Naruto University of Education, College of Education, Professor, 学校教育学部, 教授 (50127297)
ISHII Yutaka  Kyushu University, Graduate School of Mathematical Sciences, Associate Professor, 数理学研究院, 助教授 (20304727)
小澤 孝夫  龍谷大学, 理工学部, 教授 (60025913)
Project Period (FY) 2002 – 2004
Project Status Completed (Fiscal Year 2004)
Budget Amount *help
¥3,300,000 (Direct Cost: ¥3,300,000)
Fiscal Year 2004: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2003: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 2002: ¥1,500,000 (Direct Cost: ¥1,500,000)
Keywordsdynamical system / computational method / chaos / topological method / topological entropy / symbolic dynamical system / Conley Index / singularly perturbaed vector field / 大域的構造 / 特異摂動 / 周期軌道 / heteroclinic軌道 / Lyapunov数 / 計算機支援証明 / 微分方程式
Research Abstract

The purpose of this project is to develop the topological methods like Comley Index in order to study the global structure and the bifurcations of dynamical systems given by differential equations and/or mappings, moreover to realize such methos as computer algorithm. We obtained the following results during 2002 and 2004.
[A] Conley index theory for singularly perturbed vector fields and its applications----The principal idea is to use Conley index for the existence of certain solutions in singularly perturbed vector fields. A convenient formulation of such argument is given by Gedeon-Kokubu-Mischaikow-Oka-Reineck (1999), and using this, a variety of solutions described in terms of symbolic dynamics are obtained in an equations modeling fluid dynamics in a shallow container. The framework is then extended to the case of multi-dimensional slow manifold. As an application, an alternative proof for the existence of periodic traveling waves in some reaction-diffusion system studied by Gardner-Smoller (1983), and furthermore, a set of solutions described in terms of symbolic dynamics as above is also obtained there.
[B] Study of holomorphic vector fields-Aiming at the extension of Poincare-Bendixson type theorem for codimension one holomorphic foliations, several results such as non-transversality of such foliations to the boundary sphere, are obtained.
[C] Uniform hyperbolicity and non-maximal entropy locus of Henon Family-One of the results is the following. Henon mappings given by cubic polynomials at the parameter value (a, b)=(-1.35, 0.2) is not conjugate to any small perturbed mappings of 1-dimensiomal polynomial which is uniformly hyperbolic and expanding.
[D] Braid type of the set of fixed points-An equivalence relation is introduced to the set of the fixed points using notion of the braid type for orientaion preserving homeomorphisms on discs with finite fiexed points. It is shown that the braid type of the fixed points determines the fixed point index completetly.

Report

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

    (21 results)

All 2005 2004 2003 2002 Other

All Journal Article (13 results) Book (1 results) Publications (7 results)

  • [Journal Article] A Poincare-Hopf type theorem for holomorphic one-forms2005

    • Author(s)
      T.Ito, B.Scardua
    • Journal Title

      Topology 44

      Pages: 73-84

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] A Poincare-Hopf-type theorem for holmorphic one-forms2005

    • Author(s)
      T.Ito, B.Scardua
    • Journal Title

      Tepology 44

      Pages: 73-84

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Conley index theory for slow-fast systems : multi-dimensional Slow manifold2004

    • Author(s)
      岡 宏枝
    • Journal Title

      京都大学数理解析研究所講究録 1369

      Pages: 128-134

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Holomorphic foliations of codimension one transverse to polydiscs2004

    • Author(s)
      T.Ito, B.Scardua
    • Journal Title

      J.fur.die reine und angew.Math 575

      Pages: 37-44

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Conley index theory for slow-fast systems : multi-dimensional Slow manifold2004

    • Author(s)
      Hiroe Oka
    • Journal Title

      RIMS proceedings 1369

      Pages: 128-134

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Holomorphic foliations of codimension one transvers to polydiscs2004

    • Author(s)
      T.Ito, B.Scardua
    • Journal Title

      J.fur die reine und angew. Math. 575

      Pages: 37-44

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Conley Index theory for slow-fast systems : multi-dimensional Slow manifold2004

    • Author(s)
      岡 宏枝
    • Journal Title

      京都大学数理解析研究所講究録 1369

      Pages: 128-134

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Holomorphic foliation of codimension one transverse to polydiscs2004

    • Author(s)
      T.Ito, B.Scardua
    • Journal Title

      Journal for die reine undangewandte Mathematik 575

      Pages: 37-44

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Conley index theory for slow-fast systems : multi-dimensional slow manifold2003

    • Author(s)
      Hiroe Oka
    • Journal Title

      Proceedings of International Conference on Differential Equations

      Pages: 912-917

    • 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] A lap number formula in higher dimensions and rigorous entropy estimates for Lozi maps2002

    • Author(s)
      Y.Ishii, A.Shudo, K.S.Ikeda
    • Journal Title

      J.Phys.A : Math.Gen.35 17

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] A lap number formula in higher dimensions and rigorous entropy estimates for Lozi maps2002

    • Author(s)
      Y.Ishii, A.Shudo, K.S.Ikeda
    • Journal Title

      J.Phys. A : Math.Gen. 35,17

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] A survey of real transverse sections of holomorphic foliations

    • Author(s)
      T.Ito, B.Scardua
    • Journal Title

      京都大学数理解析研究所講究録 (To appear)

    • Related Report
      2004 Annual Research Report
  • [Book] 環境問題の理論と政策、第4章Lorenz方程式とカオス(寺田宏洲編著)2005

    • Author(s)
      岡宏枝
    • Publisher
      晃洋書房
    • Related Report
      2004 Annual Research Report
  • [Publications] Oka, Hiroe: "Conley index theory for slow-fast systems : multi-dimensional slow manifold"Proceedings of Equadiff 2003. (発表予定).

    • Related Report
      2003 Annual Research Report
  • [Publications] Yamagishi, Yoshikazu: "On the local convergence of Newton's method to a multiple root."J.Math.Soc.Japan. 55 4. 897-908 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Ito, Toshikazu, S.Bruno: "On holomorphic one-forms transverse to closed hypersurfaces"An.Acad.Brasil.Cienc.. 75 3. 265-269 (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] T.Matsuoka: "Fixed point index and braid invariant for fixed points of"Topology and its Applications. 12. 337-352 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] T.Ozawa: "Efficient algorithms for planar embedding of graphs with constraints in placing specified vertices on face boundaries"IEEE International symposium on circuits and systems proceedings. v949-v975 (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

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

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