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Topological indices of dynamical systems

Research Project

Project/Area Number 09440043
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionRyukoku University

Principal Investigator

YAMAGISHI Yoshikazu  Ryukoku University, Department of Applied Mathematics and Informatics, Instructor, 理工学部, 助手 (40247820)

Co-Investigator(Kenkyū-buntansha) USHIKI Shigehire  Kyoto University, Graduate School of Human and Environmental Studies, Professor, 大学院・人間環境学研究科, 教授 (10093197)
KOKUBU Hiroe (OKA)  Ryukoku University, Department of Applied Mathematics and Informatics, Professor, 理工学部, 教授 (20215221)
ITO Toshikazu  Ryukoku University, Department of Applied Mathematics and Informatics, Professor, 経済学部, 教授 (60110178)
NII Shunsaku  Saitama University, Department of Mathematics, Research Associate, 理学部, 助手 (50282421)
MATSUOKA Takashi  Naruto University of Education, College of Education, Associate Professor, 学校教育学部, 助教授 (50127297)
早川 英治郎  富山大学, 工学部, 助教授 (50240776)
森田 善久  龍谷大学, 理工学部, 教授 (10192783)
四ツ谷 晶二  龍谷大学, 理工学部, 教授 (60128361)
山口 昌哉  龍谷大学, 理工学部, 教授 (30025796)
Project Period (FY) 1997 – 1999
Project Status Completed (Fiscal Year 1999)
Budget Amount *help
¥10,500,000 (Direct Cost: ¥10,500,000)
Fiscal Year 1999: ¥3,100,000 (Direct Cost: ¥3,100,000)
Fiscal Year 1998: ¥3,100,000 (Direct Cost: ¥3,100,000)
Fiscal Year 1997: ¥4,300,000 (Direct Cost: ¥4,300,000)
Keywordstopological index / periodic indeterminate point / Ruelle's transfer operator / homoclinic orbit / fixed point index / bifurcation structure of traveling waves / surface homeomorphism / stability / Ruelleのtransfer作用素 / 周期軌道 / Conley index / ホモクリニック分岐 / connecting orbit / Thurston's canonical form / カオス / 自己 / 安定解
Research Abstract

In this project various kinds of problems which arise in dynamical systems (defied as iteration of maps or vector fields on a manifold) are studied using several types of topological methods, which resulted in the following :
1. In complex dynamical systems arising from the Newton's method for polynomial equations with 2 variables, the geometric structure around a periodic indeterminate point is investigated by constructing a Cantor set family ;of invariant manifolds.
2. Using a topological invariant of dynamical systems called Conley index, an axiomatic definition of the transition matrix and its generalization to higher parameter dimensions is given, which describes changes of the global structure of dynamical systems caused by global orbits such as homoclinic as well as heteroclinic orbits.
3. Topological structure of periodic point sets of planar maps is studied using the braid invariant and the fixed point index, and an estimate of the number of periodic points and their stability are described using these invariants.
4. Bifurcation structure of traveling waves are described in terms of a topological invariant called the stability index, and a method for determining the stability of traveling waves from the bifurcation information is obtained.
5. An improved deformation algorithm using graphs is obtained for constructing canonical forms of surface homoeomorphism. Topology of branched surfaces that admit expanding immersions is also studied.
6. A new approach for studying complex dynamical systems using Ruelle's transfer operators is developed.
7. Several applications of Poincare-Bendixson type theorems for holomorphic vector fields are obtained.
These results clearly show the effectiveness of the topological methods in the study of dynamical systems, and such an approach will become even more important in the future research.

Report

(4 results)
  • 1999 Annual Research Report   Final Research Report Summary
  • 1998 Annual Research Report
  • 1997 Annual Research Report
  • Research Products

    (35 results)

All Other

All Publications (35 results)

  • [Publications] Hiroshi Kokubu, Konstantin Mischaikow, and Hiroe Oka: "Directional transition matrix""Conley Index Theory"(Eds. K. Mischaikow, M. Mrozek, P. Zgliczynski). 47. 133-144 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] T. Gedeon, H. Kokubu, K. Mischaikow, H. Oka, J. Reineck: "Conley index for fast-slow systems I : One-dimensional slow variable"Journal of Dynamics and Differential Equations. 11. 427-470 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] S. Ushiki: "Complex Ruelle Operator in a Parabolic Basin"RIMS Kokyuroku "Research on Complex Dynamical Systems ― where it is and where it is going". 1087. 108-119 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] T. Matsuoka: "On the linking structure of periodic orbits for embeddings on the disk"to appear in Math. Japonica.

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] S. Nii: "A topological proof of stability on N-front solutions of the FitzHugh-Nagumo equations"Jounal of Dynamics and Differential Equations. 11. 515-555 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Eijirou Hayakawa: "Markov maps on trees"to appear in Math. Japonica.

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] H.Kokubu, K.Mischaikow, and H.Oka: ""Directional transition matrix," in "Conley Index Theory""(Eds. K.Mischaikow, M.Mrozek, P.Zgliczynski) Banach Center Publication. 47. 133-144 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] T.Gedeon, H.Kokubu, K.Mischaikow, H.Oka, and J.Reineck: "Conley index for fast-slow systems I : One-dimensional slow variable"Journal of Dynamics and Differential Equations. 11. 427-470 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] S.Ushiki: "Complex Ruelle Operator in a Parabolic Basin"in RIMS Kokyuroku 1087 "Research on Complex Dynamical Systems - where it is and where it is going". 108-119 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] T.Matsuoka: "On the linking structure of periodic orbits for embeddings on the disk"Mathematica Japonica. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] S.Nii: "A topological proof of stability of N-front solutions of the FitzHugh-Nagumo equations"Journal of Dynamics and Differential equations. 11. 515-555 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] E.Hayakawa: "Markov maps on trees"Mathematica Japonica. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Hiroshi Kokubu,Konstantin Mischaikow,and Hiroe Oka: "Directional transition matrix"Conley Index Theory"(Eds.K.Mischaikow,M.Mrozedk,P.Zgliczynski). 47. 133-144 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] T.Gedeon,H.Kokubu,K.Mischaikow,Hiroe Oka,and James Reineck: "Conley index for fast-slow systems I : One-dimensional slow variable"Journal of Dynamics and Differential Equations. 11. 427-470 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] S.Ushiki: "Complex Ruelle Operator in a Parabolic Basin"RIMS Kokyuroku "Research on Complex Dynamical Systems-where it is and where it is going". 1087. 108-119 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] T.Matsuoka: "On the linking structure of periodic orbits for embeddings on the disk"to appear in Math.Japonica.

    • Related Report
      1999 Annual Research Report
  • [Publications] S.Nii: "A topological proof of stability of N-front solutions of the FitzHugh-Nagumo equations"Journal of Dynamics and Differential Equations. 11. 515-555 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] Eijirou Hayakawa: "Markov maps on trees"to appear in Math. Japonica.

    • Related Report
      1999 Annual Research Report
  • [Publications] H.Kokubu,et.al.: "Connecting orbit structure of monotone solutions in the shadow system" Journal of Differential Equations. 140. 309-364 (1997)

    • Related Report
      1998 Annual Research Report
  • [Publications] T.Matsuoka: "Periodic points of disk homeomorphisms having a pseudo-Anosou component" Hokkaido Math.J.27・1. 1-33 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] E.Hayakawa: "Markov maps on trees" Math.Japonica. 51・2 (to appear).

    • Related Report
      1998 Annual Research Report
  • [Publications] Y.Morita,et.al.: "Homoclinic bifurcations in a diffusively coupled excitable system" Conference on Differential equations with applications to biology. 21. 397-407 (1999)

    • Related Report
      1998 Annual Research Report
  • [Publications] Y.Kabeya,et al.: "Number of zero of solutions to singular initial ualue problems" Tohoku Math.J.50. 1-22 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] Shunsaku Nii: "N-homoclinic bifurcations for twisted homoclinic orbits" The Proceedings of the International Conference on Dynamical Systems and Chaos,World Scientific. 206-209 (1995)

    • Related Report
      1997 Annual Research Report
  • [Publications] Shunsaku Nii: "N-homoclinic bifurcations for homoclinic orbits changing their twisting" Jounal of Dynamics and Differential Equations. 8. 549-572 (1996)

    • Related Report
      1997 Annual Research Report
  • [Publications] Shunsaku Nii: "An extension of the stability index for travelling wave solutions and its application for bifurcations" SIAM Journal on Mathematical Analysis. 28. 402-433 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] Shunsaku Nii: "Stability of travelling multiple-front (multiple-back) wave solutions of the Fitz Hugh-Nagumo equations" SIAM Journal on Mathematical Analysis. 28. 1094-1112 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] Shunsaku Nii: "進行波に沿った線型化固有値問題へのTopologicalなアプローチ" 数理解析研究所講究録. 969. 18-28 (1996)

    • Related Report
      1997 Annual Research Report
  • [Publications] Y.Yamagishi: "stable spiralorbits SoR Purand-Kerner's method applied to the equation X^d=0" Journal of Computational and Applied Mathematics. 82. 465-475 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] T.Matsuoka: "Periodic points of disk homeomorphisms having a pseudo-Anosov component" to appear in Hokkaido Math.J.(1997).

    • Related Report
      1997 Annual Research Report
  • [Publications] 山口昌哉: "複雑系科学の哲学に不足しているもの" 日本ファジ-学会誌. 9・5. 603-613 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] Y.Kabeya, E.Yanagida and S.Yotsutani: "Number of zeros of solutions to singular initial value problems" to appear in Tohoku J.Math.

    • Related Report
      1997 Annual Research Report
  • [Publications] X-Y.Chen, S.Jimbo and Y.Morita: "Stabilization of Vortices in the Ginzburg-Landau Equation with a Variable Diffusion Coefficient" SIAM Journal of Mathematical Analysis,To appear.

    • Related Report
      1997 Annual Research Report
  • [Publications] Y.Morita: "Stability of Solutions to the Ginzburg-Landau Equation with Neumann Condition" Nonlinear Analysis: Theory,Methods and Applications. 30・6. 3939-3946 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] Masaya Yamaguti, Masayoshi Hata, Jun Kigami: "Translations of Mathematical Monographs,vol.167" American Mathematical Society, 78 (1997)

    • Related Report
      1997 Annual Research Report

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Published: 1997-04-01   Modified: 2016-11-11  

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