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Bifurcations of vector fields possessing the shadowing property

Research Project

Project/Area Number 13640225
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

SAKAI Kazuhiro  Utsunomiya Univ., Dept. of Education, Associate Prof., 教育学部, 助教授 (30205702)

Co-Investigator(Kenkyū-buntansha) HOSAKA Tetsuya  Utsunomiya Univ., Dept. of Education, Lecturer, 教育学部, 講師 (50344908)
KITAGAWA Yoshihisa  Utsunomiya Univ., Dept. of Education, Professor, 教育学部, 教授 (20144917)
FUJIHIRA Hideyuki  Utsunomiya Univ., Dept. of Education, Professor, 教育学部, 教授 (70114171)
木村 寛  宇都宮大学, 教育学部, 教授 (70017953)
落合 昭二  宇都宮大学, 教育学部, 教授 (30031545)
江森 英世  宇都宮大学, 教育学部, 助教授 (90267526)
Project Period (FY) 2001 – 2002
Project Status Completed (Fiscal Year 2002)
Budget Amount *help
¥2,300,000 (Direct Cost: ¥2,300,000)
Fiscal Year 2002: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 2001: ¥1,300,000 (Direct Cost: ¥1,300,000)
Keywordsdynamical systems theory / bifurcation / shadowing property / Axiom A / strong transversality condition / pseudo orbits / vector fields / カオス / connectin lemma
Research Abstract

The purpose of this research project is to analyze the bifurcation phenomena of a 1-parameter family, which contains a vector field possessing the shadowing property (abbr.SP), composed of differentiate vector fields.
Denote by SP the set of vector fields possessing SP endowed with the C^1-topology, and let intSP be the C^1-interior. At the first stage of this research, it was indispensable to characterize the vector fields in intSP by making use of the stability theory of dynamical systems. In the end of 2001, however, we could not get a complete characterization for the vector fields.
In 2002, we continuously proceeded to characterize the above vector fields, and to analyze the bifurcation phenomena of such 1-parameter family. The main obstacle to the characterization is the existence of singularities of X ∈ intSP. We intimately analyzed the behavior of the orbits of the integrated flow X_t in the neighborhood of singularities, and we have found a special (but very natural) property of the flow being displayed in the neighborhood. Let us call the property (*) for simplicity. Very recently, it has been proved that X ∈ intSP satisfies (*) if and only if X satisfies both Axiom A and the strong transversality condition. This result is a partial answer to the problem we aimed to solve in 2001 at the first stage. Nowadays, using the above result we are analyzing the bifurcation phenomena of a 1-parameter family which contains a vector field possessing SP. In our opinion, the achieve percentage of this research project might be evaluated 70 %.

Report

(3 results)
  • 2002 Annual Research Report   Final Research Report Summary
  • 2001 Annual Research Report
  • Research Products

    (49 results)

All Other

All Publications (49 results)

  • [Publications] Pilyugin S.Yu., Rodionova, A.A., Sakai, K.: "Orbital and weak shadowing properties"Discrete and Continuous Dynamical Systems. 9. 287-308 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Sakai, K.: "Diffeomorphisms with C^2-stable shadowing"Dynamical Systems : An International Journal. 17. 235-241 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Sakai, K.: "A note on weak shadowing"Far East Journal of Dynamical Systems. 3. 45-49 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Sakai, K.: "Diffeomorphisms with weak shadowing"Fundamenta Mathematicae. 168. 57-75 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Moriyasu, K., Sakai, K., Sumi, N.: "Vector fields with topological stability"Transactions of the American Mathematical Society. 353. 3391-3408 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Sakai, K.: "Shadowing properties of L-hyperbolic homeomorphisms"Topology and its Applications. 112. 229-243 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Sakai, K.: "Various shadowing properties for positively expansive maps"Topology and its Applications. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Moriyasu, K., Sakai, K.: "A note on Mane's proof of the stability conjecture"Far East Journal of Dynamical Systems. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Kitagawa, Y.: "Deformable flat tori in S^3 with constant mean curvature"Osaka Journal of Mathematics. 40. 103-119 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Hosaka, T.: "Limit sets of geometrically finite groups acting on Busemann spaces"Topology and its Applications. 122. 565-580 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Hosaka, T.: "Parabolic subgroups of finite index in Coxeter groups"Journal of Pure and Applied Algebra. 169. 215-227 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Hosaka, T.: "On the cohomology of Coxeter groups"Journal of Pure and Applied Algebra. 162. 291-301 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Hosaka, T.: "A splitting theorem for CAT(0) spaces with the geodesic extension property"Tsukuba Journal of Mathematics. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Hosaka, T.: "The interior of the limit set of groups"Houston Journal of Mathematics. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] S. Yu. Pilyugin, A. A. Rodionova and K. Sakai: "Orbital and weak shadowing properties"Discrete and Continuous Dynamical Systems. 9. 287-308 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] K. Sakai: "Diffeomorphisms with C^2 stable shadowing"Dynamical Systems : An International Journal. 17. 235-241 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] K. Sakai: "A note on weak shadowing"Far East Journal of Dynamical Systems. 3. 45-49 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] K. Sakai: "Diffeomorphisms with weak shadowing"Fundamenta Mathematicae. 168. 57-75 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] K. Moriyasu, K. Sakai and N. Sumi: "Vector fields with topological stability"Transactions of the American Mathematical Society. 353. 3391-3408 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] K. Sakai: "Shadowing properties of L-hyperbolic homeomorphisms"Topology and its Applications. 112. 229-243 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] K. Sakai: "Various shadowing properties for positively expansive maps"Topology and its Applications. to appear.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] K. Moriyasu and K. Sakai: "A note on Mane's proof of the stability conjecture"Far East Journal of Dynamical Systems. to appear.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Y. Kitagawa: "Deformable flat tori in S^3 with constant mean curvature"Osaka Journal of Mathematics. 40. 103-119 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] T. Hosaka: "Limit sets of geometrically finite groups acting on Busemann spaces"Topology and its Applications. 122. 565-580 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] T. Hosaka: "Parabolic subgroups of finite index in Coxeter groups"Journal of Pure and Applied Algebra. 169. 215-227 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] T. Hosaka: "On the cohomology of Coxeter groups"Journal of Pure and Applied Algebra. 162. 291-301 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] T. Hosaka: "A splitting theorem for CAT(0) spaces with the geodesic extension property"Tsukuba Journal of Mathematics. to appear.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] T. Hosaka: "The interior of the limit set of group"Houston Journal of Mathematics. to appear.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Pilyugin S.Yu., Rodionova, A.A., Sakai, K.: "Orbital and weak shadowing properties"Discrete and Continuous Dynamical Systems. 9. 287-308 (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] Sakai, K.: "Diffeomorphisms with C^2-stable shadowing"Dynamical Systems : An International Journal. 17. 235-241 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Sakai, K.: "A note on weak shadowing"Far East Journal of Dynamical Systems. 3. 45-49 (2001)

    • Related Report
      2002 Annual Research Report
  • [Publications] Sakai, K.: "Diffeomorphisms with weak shadowing"Fundamenta Mathematicae. 168. 57-75 (2001)

    • Related Report
      2002 Annual Research Report
  • [Publications] Moriyasu, K., Sakai, K., Sumi, N.: "Vector fields with topological stability"Transactions of the American Mathematical Society. 353. 3391-3408 (2001)

    • Related Report
      2002 Annual Research Report
  • [Publications] Sakai, K.: "Shadowing properties of L-hyperbolic homeomorphisms"Topology and its Applications. 112. 229-243 (2001)

    • Related Report
      2002 Annual Research Report
  • [Publications] Sakai, K.: "Various shadowing properties for positively expansive maps"Topology and its Applications. (to appear).

    • Related Report
      2002 Annual Research Report
  • [Publications] Moriyasu, K., Sakai, K.: "A note on Mane's proof of the stability conjecture"Far East Journal of Dynamical Systems. (to appear).

    • Related Report
      2002 Annual Research Report
  • [Publications] Kitagawa, Y.: "Deformable flat tori in S^3 with constant mean curvature"Osaka Journal of Mathematics. 40. 103-119 (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] Hosaka, T.: "Limit sets of geometrically finite groups acting on Busemann spaces"Topology and its Applications. 122. 565-580 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Hosaka, T.: "Parabolic subgroups of finite index in Coxeter groups"Journal of Pure and Applied Algebra. 169. 215-227 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Hosaka, T.: "On the cohomology of Coxeter groups"Journal of Pure and Applied Algebra. 162. 291-301 (2001)

    • Related Report
      2002 Annual Research Report
  • [Publications] Hosaka, T.: "A splitting theorem for CAT(O) spaces with the geodesic extension property"Tsukuba Journal of Mathematics. (to appear).

    • Related Report
      2002 Annual Research Report
  • [Publications] Hosaka, T.: "The interior of the limit set of groups"Houston Journal of Mathematics. (to appear).

    • Related Report
      2002 Annual Research Report
  • [Publications] Sakai, Kazuhiro: "Diffeomorphisms with weak shadowing"Fundamenta Mathematicae. 168. 57-75 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Sakai, Kazuhiro: "Shadowing properties of {\cal L}-hyperbolic homeomorphisms"Topology and its Applications. 112. 229-243 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] K. Sakai, K. Moriyasu, N. Sumi: "Vector fields with topological stability"Transactions of the American Mathematical Society. 353. 3391-3408 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Sakai, Kazuhiro: "A note on weak shadowing"Far East Journal of Dynamical Systems. 3. 45-49 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Kitagawa, Yoshihisa: "Deformable flat tori in S^3 with constant mean curvature"Osaka Journal of Mathematics. (to appear). (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] 木村 寛: "算数・数学科における基礎・基本を再考するにあたって"下野教育(栃木県連合教育会機関誌). 703. 4-8 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] 木村 寛 編著: "算数・数学科の基礎・基本 : 小・中・高校を通して"栃木県教育研究所紀要、第121集. 110 (2001)

    • Related Report
      2001 Annual Research Report

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

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