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Chaos control associated to topoiogical dynamics

Research Project

Project/Area Number 11554001
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section展開研究
Research Field Geometry
Research InstitutionHIROSHIMA UNIVERSITY

Principal Investigator

YOSHIDA Toshio (2002)  Hiroshima University, Faculty of Integrated Arts and Sciences, Professor, 総合科学部, 教授 (10033854)

中山 裕道 (1999-2001)  広島大学, 総合科学部, 助教授 (30227970)

Co-Investigator(Kenkyū-buntansha) 吉田 敏男  広島大学, 総合科学部, 教授 (10033854)
Project Period (FY) 1999 – 2002
Project Status Completed (Fiscal Year 2002)
Budget Amount *help
¥3,600,000 (Direct Cost: ¥3,600,000)
Fiscal Year 2002: ¥700,000 (Direct Cost: ¥700,000)
Fiscal Year 2001: ¥700,000 (Direct Cost: ¥700,000)
Fiscal Year 2000: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 1999: ¥1,100,000 (Direct Cost: ¥1,100,000)
Keywordstopological dynamics / chaos control / ローレンツ水車 / ルエル不変量 / 回転数
Research Abstract

Chaotic phenomena are, roughly speaking, unpredictable ones. These subjects are examined by various simulations, but their theory is not easy to be used practically. Recently Ott, Grebogi and Yorke proposed the chaos control, which has a possibility of practical use. In this study, we tried to formulate the chaos control mathematically and to analyze it from both mathematical and practical points of view.
As for mathematical study, we considered chaos phenomena in term of infinitesimal behavior related to Lyapunov exponent, and studied fiberwise invariant measures, Ruelle invariants, lifts of protective flows, branch points of projectively Anosov flows, chain recurrent sets of protective flows, exceptional minimal sets of codimension two. By projectivizing the othogonal projection for the derivative of a flow, we obtain a flow of a fiber bundle whose fiber is homeomorphic to the projective space. This flow is called a projective flow. Lyapunov exponent represents the infinitesimal dilatation. The projective flow means the infinitesimal twist along the orbits. By these results, Ruelle invariant, the characteristic representing the infinitesimal twist, is valid for the determination of chaotic behavior as well as Lyapunov invariant. Ruelle invariant is easy to be treated in mathematical sense. Thus they seem to be valuable.
As for applications of mathematical theory to real phenomena, we formulated mathematically multi-agents systems, switching arrival systems, chaotic mills and atmospheric phenomena. For example, for chaotic mills, we derived Markus mills from them, and found that their chaotic behaviors come from Parry maps. Parry maps are mathematically defined and their chaos control can be naturally formulated. In this point of view, we stepped out for the theoretical analysis of the chaotic control.

Report

(5 results)
  • 2002 Annual Research Report   Final Research Report Summary
  • 2001 Annual Research Report
  • 2000 Annual Research Report
  • 1999 Annual Research Report
  • Research Products

    (34 results)

All Other

All Publications (34 results)

  • [Publications] T.Kobayashi: "Stably extendible vector bundles over the real projective spaces and the lens spaces"Hiroshima Math. J.. 29-3. 631-638 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] T.Kobayashi: "Stable extendibility of normal bundles associated to immersions of real projective spaces and lens spaces"Mem. Fac. Sci. Kochi Univ. (Math.). 21. 31-38 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] T.Kobayashi: "Extendibility and stable extendibility of the power of the normal bundle associated to an immersion of the lens space mod 4"Mem. Fac. Sci. Kochi Univ. (Math.). 22. 45-57 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] H.Nakayama: "Transitively twisted flows of 3-manifolds"Commentarii Mathematical Helvetici. 76-4. 577-588 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] S.Matsumoto: "On the Ruelle invariants for diffeomorphisms of the torus"Ergodic Theory and Dyanmical Systems. 22. 1263-1267 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] T.Kobayashi: "Extendibility, stable extendibility and span of vector bundles mξ_n over real projective spaces"Adv. Stud. Contemp. Math.. 5-2. 189-199 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] T.Kobayashi: "Extendibility and stable extendibility of normal bundles associated to immersions of real projective spaces"Osaka J. Math.. 39-2. 315-324 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] T.Kobayashi: "Extendibility, stable extendibility and span of some vector bundles over real projective spaces"Mem. Fac. Sci. Kochi Univ. (Math.). 23. 45-56 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] T.Kobayashi: "Extendible and stably extendible vector bundles over real projective spaces"J. Math. Soc. Japan. 55-4. (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] T.Kobayashi: "Stable extendibility of vector bundles over real projective spaces and determination of the Schwarzenberger number β(k)"Mem. Fac. Sci. Kochi Univ. (Math.). 24. 19-35 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] T.Inaba: "Invariant fiber measures of angular flows and the Ruelle invariant"J. Math. Soc. Japan. 55. (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] T. Kobayashi: "Stably extendible vector bundles over the real protective spaces and the lens spaces"Hiroshima Math. J.. 29-3. 631-638 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] T. Kobayashi: "Stable extendibility of normal bundles associated to immersions of real projective spaces and lens spaces"Mem. Fac, Sci. Kochi Univ. (Math.). 21. 31-38 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] T. Kobayaishi: "Extendibiiity and stable extendibiiity of the power of the normal bundle associated to an immersion of the lens space mod 4"Mem. Fac. Sci. Kochi Univ. (Math.). 22. 45-57 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] H. Nakayama: "Transitively twisted flows of 3-manifolds"Commentarii Mathematical Helvetici. 76-4. 577-588 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] S. Matsumoto: "On the Ruelle invariants for diffeomorphisms of the torus"Ergodic Theory and Dyanmical Systems. 22. 1263-1267 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] T. Kobayashi: "Extendibiiity, stable extendibiiity and span of vector bundles mξ_n over real projective spaces"Adv. Stud. Contemp. Math.. 5-2. 189-199 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] T. Kobayashi: "Extendibiiity and stable extendibiiity of normal bundles associated to immersions of real projective spaces"Osaka J. Math.. 39-2. 315-324 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] T. Kobayashi: "Extendibiiity, stable extendibiiity and span of some vector bundles over real projective spaces"Mem. Fac. Sci; Kochi Univ. (Math.). 23. 45-56 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] T. Kobayashi: "Extendible and stably extendible vector bundles over real projective spaces"J. Math. Soc. Japan. 55-4. (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] T. Kobayashi: "Stable extendibiiity of vector bundles over real projective spaces and determination of the Schwarzenberger number β (k)"Mem. Fac. Sci. Kochi Univ. (Math.). 24. 19-35 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] T. Inaba: "Invariant fiber measures of angular flows and the Ruelle invariant"J. Math, Soc. Japan. 55. (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] 小林貞一, 吉田敏男: "Extendible and stably extendible vector bundles over real projective spaces"J. Math. Soc. Japan. 55巻4号(印刷中). (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] 小林貞一, 吉田敏男: "Stable extendibility of vector bundles over real projective spaces and determination of the Schwarzenberger number β (k)"Mem. Fac. Sci. Kochi Univ. (Math.). 24巻. 19-35 (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] 小林貞一, 吉田敏男: "Extendibility, stable extendibility and span of vector bundles m ξ n over real projective spaces"Adv. Stud. Contemp. Math.. 5巻2号. 189-199 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] 小林貞一, 牧春夫, 吉田敏男: "Extendibility and stable extendibility of normal bundles associated to immersions of real projective spaces"0saka J. Math.. 39巻2号. 315-324 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] 小林貞一, 吉田敏男: "Extendibity, stable extendibility and span of some vector bundles over real projective spaces"Mem. Fac. Sci. Kochi Univ. (Math.). 23巻. 45-56 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] H.Nakayama: "Transitively twisted flows of 3-manifolds"Commentarii Mathematici Helvetici. 76-4. 577-588 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] S.Matsumoto: "On the Ruelle invariants for diffeomorphisms of the two torus"Ergodic Theory and Dynamical Systems. (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] T.Kobayashi: "Extendibility, stable extendibility and span of some vector bundles over real projective spaces"Mem. Fac. Sci. Kochi Univ.(Math.). 23. 45-56 (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] T.Kobayashi: "Extendibility and stable extendibility of normal bundles associated to immersions of real projective spaces"Osaka J. of Math.. (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Teiichi Kobayashi,Haruo Maki and Toshio Yoshida: "Extendibility and stable extendibility of the power of the normal bundle associated to an immersion of the lens space mod 4"Mem.Fac.Sci.Kochi Univ.(Math.). 22. 45-57 (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] Teiichi Kobayashi: "Stably extendible vector bundles over the real projective spaces and the lens spaces"Hiroshima Math. J.. 29. 631-638 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] Teiichi Kobayashi: "Stable extendibility of normal bundles associated to immersions of real projective spaces and lens spaces"Mem. Fac. Sci. Kochi Univ. Ser. A Math.. 21. 31-38 (2000)

    • Related Report
      1999 Annual Research Report

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

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