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Study of Global Structures and Bifurcations of Dynamical Systems including Systems with Large Degrees of Freedom

Research Project

Project/Area Number 21340035
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  京都大学, 大学院・理学研究科, 教授 (50202057)

Co-Investigator(Renkei-kenkyūsha) ARAI Zin  北海道大学, 大学院・理学研究院, 准教授 (80362432)
OKA Hiroe  龍谷大学, 理工学部, 教授 (20215221)
OBAYASHI Ippei  京都大学, 大学院・理学研究科, 特定研究員 (30583455)
Project Period (FY) 2009 – 2012
Project Status Completed (Fiscal Year 2012)
Budget Amount *help
¥17,160,000 (Direct Cost: ¥13,200,000、Indirect Cost: ¥3,960,000)
Fiscal Year 2012: ¥3,510,000 (Direct Cost: ¥2,700,000、Indirect Cost: ¥810,000)
Fiscal Year 2011: ¥3,900,000 (Direct Cost: ¥3,000,000、Indirect Cost: ¥900,000)
Fiscal Year 2010: ¥4,030,000 (Direct Cost: ¥3,100,000、Indirect Cost: ¥930,000)
Fiscal Year 2009: ¥5,720,000 (Direct Cost: ¥4,400,000、Indirect Cost: ¥1,320,000)
Keywords力学系 / 分岐 / 大域的構造 / 不変集合 / 計算機援用証明 / 位相計算的方法 / 大自由度系 / 計算機援用解析 / カオス / Morse分解 / グラフアルゴリズム / 計算機援用 / 位相計算 / 精度保証付き計 / ホモロジー / グラフ・アルゴリズム
Research Abstract

A great improvement of the Conley-Morse graph method, a computer-assisted method that analyzes global dynamics and bifurcations by combining a topological method and validated numerical computation has been done by developing a computer-algorithm for generating so-called clutching graphs which describe relation between Conley-Morse graphs on adjacent parameter domains. Another improvement is a method of non-uniform grid decomposition of the phase space in order to substantially decrease computational cost of the Conley-Morse graph method.A new approach to the bifurcation theory of dynamical systems from topological-computation theory viewpoint has been proposed which captures bifurcation of dynamics from changes of the corresponding Conley-Morse graphs. From this viewpoint, we have obtained a new mathematical framework for the crisis bifurcation.Moreover, we have studied the global dynamics of a Coupled Map Lattice system and a Coupled Oscillator system by using the Conley-Morse graph method, and have obtained the creation and bifurcations of some global dynamics such as unstable invariant tori and their connecting orbits.

Report

(5 results)
  • 2012 Annual Research Report   Final Research Report ( PDF )
  • 2011 Annual Research Report
  • 2010 Annual Research Report
  • 2009 Annual Research Report
  • Research Products

    (44 results)

All 2013 2012 2011 2010 2009

All Journal Article (11 results) (of which Peer Reviewed: 9 results) Presentation (32 results) (of which Invited: 4 results) Book (1 results)

  • [Journal Article] A topological computation approach to the interior crisis bifurcation2013

    • Author(s)
      Hiroshi Kokubu and Hiroe Oka
    • Journal Title

      Nonlinear Theory and Its Applications, IEICE

      Volume: 4 Issue: 1 Pages: 97-103

    • DOI

      10.1587/nolta.4.97

    • NAID

      130003375415

    • ISSN
      2185-4106
    • Related Report
      2012 Annual Research Report 2012 Final Research Report
    • Peer Reviewed
  • [Journal Article] Graph-based topological approximation of saddle-node bifurcation in maps2012

    • Author(s)
      Zin Arai, Marcio Gameiro, Tomas Gedeon, Hiroshi Kokubu, Konstantin Mischaikow, Hiroe Oka
    • Journal Title

      RIMS Kokyuroku Bessatsu B31

      Pages: 225-241

    • Related Report
      2012 Final Research Report
    • Peer Reviewed
  • [Journal Article] Combinatorial-topological framework for the analysis of global dynamics2012

    • Author(s)
      Justin Bush, Marcio Gameiro, Shaun Harker, Hiroshi Kokubu, Konstantin Mischaikow, Ippei Obayashi, Pawel Pilarczyk
    • Journal Title

      Chaos

      Volume: 22 Issue: 4 Pages: 047508-047508

    • DOI

      10.1063/1.4767672

    • Related Report
      2012 Annual Research Report 2012 Final Research Report
    • Peer Reviewed
  • [Journal Article] A combinatorial framework for analysis of global dynamics and bifurcations2012

    • Author(s)
      Hiroshi Kokubu and Konstantin Mischaikow
    • Journal Title

      IUTAM Symposium on 50 Years of Chaos: Applied and Theoretical Procedia IUTAM

      Volume: 5 Pages: 195-198

    • Related Report
      2012 Final Research Report
  • [Journal Article] A combinatorial framework for analysis of global dynamics and bifurcations2012

    • Author(s)
      Hiroshi Kokubu and Konstantin Mischaikow
    • Journal Title

      Procedia IUTAM

      Volume: 5 Pages: 195-198

    • Related Report
      2012 Annual Research Report
  • [Journal Article] Graph-based topological approximation of saddle-node bifurcation in maps2012

    • Author(s)
      Zin Arai, Marcio Gameiro, Tomas Gedeon, Hiroshi Kokubu, Konstantin Mischaikow, Hiroe Oka
    • Journal Title

      RIMS Kokyuroku Bessatsu

      Volume: B31 Pages: 225-241

    • Related Report
      2011 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Capturing the global behavior of dynamical systems with Conley-Morse graphs2012

    • Author(s)
      Z.Arai, H.Kokubu, I.Obayashi
    • Journal Title

      Advances in Cognitive Neurodynamics (III)

      Volume: (掲載確定)(未定)

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Recent development in rigorous computational methods in dynamical systems2009

    • Author(s)
      Zin Arai, Hiroshi Kokubu, and Pawel Pilarczyk
    • Journal Title

      Japan Journal of Industrial and Applied Mathematics

      Volume: 26 Pages: 393-417

    • NAID

      10028169214

    • Related Report
      2012 Final Research Report
    • Peer Reviewed
  • [Journal Article] A database schema for the analysis of global dynamics of multiparameter systems2009

    • Author(s)
      Zin Arai, William Kalies, Hiroshi Kokubu, Konstantin Mischaikow, Hiroe Oka, and Pawel Pilarczyk
    • Journal Title

      SIAM Journal on Applied Dynamical Systems

      Volume: 8 Pages: 757-789

    • NAID

      120001629760

    • Related Report
      2012 Final Research Report
    • Peer Reviewed
  • [Journal Article] A database schema for the analysis of global dynamics of multiparameter systems2009

    • Author(s)
      Z.Arai, W.Kalies, H.Kokubu, K.Mischaikow, H.Oka, P.Pilarczyk
    • Journal Title

      SIAM Journal on Applied Dynamical Systems 8

      Pages: 757-789

    • NAID

      120001629760

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Recent development in rigorous computational methods in dynamical systems2009

    • Author(s)
      Zin Arai, Hiroshi Kokubu, Pawel Pilarczyk
    • Journal Title

      Japan Journal of Industrial and Applied Mathematics 26

      Pages: 393-417

    • NAID

      10028169214

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Presentation] Topological Computation Method for Dynamical Time Series Analysis and its Applications2013

    • Author(s)
      Hiroshi Kokubu
    • Organizer
      The First International Conference on Dynamics of Differential Equations
    • Place of Presentation
      Georgia Institute of Technology,Atlanta, U.S.A
    • Related Report
      2012 Final Research Report
  • [Presentation] ダイナミクス全構造計算による新しい相空間構造解析の方法とその応用2012

    • Author(s)
      國府寛司
    • Organizer
      計測自動制御学会 第50回自律分散システム部会研究会
    • Place of Presentation
      京都大学吉田キャンパス 京都京都大学吉田キャンパス 京都大学百周年時計台記念館 国際交流ホール
    • Year and Date
      2012-06-15
    • Related Report
      2012 Final Research Report
  • [Presentation] Topological computation method for global dynamics and its application2012

    • Author(s)
      Hiroshi Kokubu
    • Organizer
      Nonlinear Partial Differential Equations
    • Place of Presentation
      DynamicalSystems and Their Applications, RIMS, Kyoto University
    • Related Report
      2012 Final Research Report
  • [Presentation] Topological Computation Methods for Global Dynamics and Bifurcations2012

    • Author(s)
      Hiroshi Kokubu
    • Organizer
      Workshop on Dynamical Systems
    • Place of Presentation
      National Center for Theoretical Science, National Tsinghua University, Hsinchu
    • Related Report
      2012 Final Research Report
  • [Presentation] Topological Computation Methods for Global Dynamics and Bifurcations Part 1: Theory, Part 2: Application2012

    • Author(s)
      Hiroshi Kokubu
    • Organizer
      NCTS Workshop on Dynamical Systems
    • Place of Presentation
      NCTS, National Tsinghua University Hsinchu, Taiwan
    • Related Report
      2012 Annual Research Report
    • Invited
  • [Presentation] ダイナミクス全構造計算による新しい相空間構造解析の方法とその応用2012

    • Author(s)
      國府寛司
    • Organizer
      計測自動制御学会 第50回自律分散システム部会研究会
    • Place of Presentation
      京都大学百周年時計台記念館
    • Related Report
      2012 Annual Research Report
    • Invited
  • [Presentation] Topological computation method for global dynamics and its application2012

    • Author(s)
      Hiroshi Kokubu
    • Organizer
      Nonlinear Partial Differential Equations, Dynamical Systems and Their Applications
    • Place of Presentation
      RIMS, Kyoto University
    • Related Report
      2012 Annual Research Report
    • Invited
  • [Presentation] Time series analysis using topological computation method: a preliminary report2012

    • Author(s)
      Hiroshi Kokubu
    • Organizer
      ICMC Summer Meeting on Differential Equations
    • Place of Presentation
      Sao Carlos, Brazil
    • Related Report
      2011 Annual Research Report
  • [Presentation] Topological bifurcation theorems for Morse decompositions2011

    • Author(s)
      Hiroshi Kokubu
    • Organizer
      International Conference "Far-From-Equilibrium Dynamics"
    • Place of Presentation
      京都大学数理解析研究所(招待講演)
    • Year and Date
      2011-01-08
    • Related Report
      2010 Annual Research Report
  • [Presentation] Topological-computational methods for analyzing global dynamics and bifurcations2011

    • Author(s)
      Hiroshi Kokubu
    • Organizer
      IUTAM Symposium on 50 years of chaos
    • Place of Presentation
      Kyoto University, Japan
    • Related Report
      2012 Final Research Report
  • [Presentation] Graph-based topological computation method for global analysis of dynamics and its application to time series analysis2011

    • Author(s)
      Hiroshi Kokubu
    • Organizer
      Workshop on Applied Topology
    • Place of Presentation
      JR博多駅ビル10F 博多シティ会議室
    • Related Report
      2012 Final Research Report
  • [Presentation] Computational Analysis of global dynamics and bifurcations2011

    • Author(s)
      Hiroshi Kokubu
    • Organizer
      Mini-Symposium "Computational Approach to Global Dynamics"
    • Place of Presentation
      ICIAM, Vancouver, Canada
    • Related Report
      2012 Final Research Report
  • [Presentation] Graph based topological computation for global analysis of dynamical systems2011

    • Author(s)
      Hiroshi Kokubu
    • Organizer
      Conference on Computational Methods in Dynamics
    • Place of Presentation
      ICTP, Trieste, Italy
    • Related Report
      2012 Final Research Report 2011 Annual Research Report
  • [Presentation] Topological bifurcation theorems for Morse decompositions2011

    • Author(s)
      Hiroshi Kokubu
    • Organizer
      International Conference "Far-From-Equilibrium Dynamics"
    • Place of Presentation
      RIMS, Kyoto University, Kyoto, Japan
    • Related Report
      2012 Final Research Report
  • [Presentation] Application of topological computation method for global dynamics of an associative memory model of the Hopfield type2011

    • Author(s)
      Hiroshi Kokubu
    • Organizer
      East Asia SIAM 2011
    • Place of Presentation
      Waseda University Kitakyushu Campus
    • Related Report
      2011 Annual Research Report
  • [Presentation] Graph-based topological computation method for global analysis of dynamics and its application to time series analysis2011

    • Author(s)
      Hiroshi Kokubu
    • Organizer
      研究集会「応用トポロジー:情報通信・生命科学との連携を目指して」
    • Place of Presentation
      JR博多駅ビル10F 博多シティ会議室
    • Related Report
      2011 Annual Research Report
  • [Presentation] Topological-computational methods for analyzing global dynamics and bifurcations2011

    • Author(s)
      Hiroshi Kokubu
    • Organizer
      IUTAM Symposium on 50 years of chaos
    • Place of Presentation
      Kyoto University
    • Related Report
      2011 Annual Research Report
    • Invited
  • [Presentation] 力学系の位相計算的方法とその応用2010

    • Author(s)
      Hiroshi Kokubu
    • Organizer
      理論生物学セミナー
    • Place of Presentation
      理化学研究所基幹研究所、和光市
    • Year and Date
      2010-11-27
    • Related Report
      2010 Annual Research Report
  • [Presentation] 力学系の位相計算的方法とその応用2010

    • Author(s)
      國府寛司
    • Organizer
      公開シンポジウム"統合複雑系科学への招待"
    • Place of Presentation
      京都大学理学研究科セミナーハウス,京都
    • Year and Date
      2010-08-05
    • Related Report
      2012 Final Research Report
  • [Presentation] 力学系の位相計算的方法とその応用2010

    • Author(s)
      Hiroshi Kokubu
    • Organizer
      公開シンポジウム"統合複雑系科学への招待"
    • Place of Presentation
      京都大学理学研究科セミナーハウス
    • Year and Date
      2010-08-05
    • Related Report
      2010 Annual Research Report
  • [Presentation] 力学系の位相計算的方法とその応用2010

    • Author(s)
      國府寛司
    • Organizer
      第9回機械工学における力学系理論の応用に関する研究会
    • Place of Presentation
      慶應義塾大学理工学部
    • Year and Date
      2010-03-28
    • Related Report
      2009 Annual Research Report
  • [Presentation] A topological-computational method for global analysis of dynamical systems2010

    • Author(s)
      Hiroshi Kokubu
    • Organizer
      RIMS Workshop "Structures of Attractors in Dissipative Systems"
    • Place of Presentation
      RIMS, Kyoto University, Kyoto, Japan
    • Related Report
      2012 Final Research Report
  • [Presentation] Rigorous Topological Computation of Dynamics and Applications2010

    • Author(s)
      Hiroshi Kokubu, Rigorous Topological Computation of Dynamics and Applications
    • Organizer
      Long-term workshop: Mathematical Sciences and Their Applications
    • Place of Presentation
      Hotel Rako-Hananoi, Kamisuwa, Nagano, Japan
    • Related Report
      2012 Final Research Report
  • [Presentation] 力学系の位相計算的方法とその応用2010

    • Author(s)
      國府寛司
    • Organizer
      力学系研究会
    • Place of Presentation
      慶應義塾大学
    • Related Report
      2012 Final Research Report
  • [Presentation] Dynamics of Hebbian Models of Associative Memory2010

    • Author(s)
      Hiroshi Kokubu
    • Organizer
      Workshop "Computational Homology & Dynamics"
    • Place of Presentation
      University of Canterbury, Christchurch, New Zeeland
    • Related Report
      2010 Annual Research Report
  • [Presentation] Rigorous Topological Computation of Dynamics and Applications2010

    • Author(s)
      Hiroshi Kokubu
    • Organizer
      Long-term workshop : Mathematical Sciences and Their Applications
    • Place of Presentation
      Hotel Rako-Hananoi,上諏訪(招待講演)
    • Related Report
      2010 Annual Research Report
  • [Presentation] A topological-computational method for global analysis of dynamical systems2010

    • Author(s)
      Hiroshi Kokubu
    • Organizer
      RIMS Workshop "Structures of Attractors in Dissipative Systems"
    • Place of Presentation
      京都大学数理解析研究所(招待講演)
    • Related Report
      2010 Annual Research Report
  • [Presentation] A method for constructing databases for global dynamics of multi-parameter systems2009

    • Author(s)
      國府寛司
    • Organizer
      Japan-Taiwan Joint Workshop on "Numerical Analysis and Scientific Computations"
    • Place of Presentation
      National Taiwan University, Taipei
    • Year and Date
      2009-11-08
    • Related Report
      2009 Annual Research Report
  • [Presentation] A method for constructing databases for global dynamics of multi-parameter systems2009

    • Author(s)
      國府寛司
    • Organizer
      International Workshop on "What is Evolution?" -Bicentennial of Charles Darwin's Birth-
    • Place of Presentation
      COOP-INN Kyoto Conference Hall
    • Year and Date
      2009-10-16
    • Related Report
      2009 Annual Research Report
  • [Presentation] A method for constructing databases for global dynamics of multi-parameter systems2009

    • Author(s)
      Hiroshi Kokubu
    • Organizer
      Japan-Taiwan Joint Workshop on "Numerical Analysis and Scientific Computations"
    • Place of Presentation
      Department of Mathematics, National Taiwan University
    • Related Report
      2012 Final Research Report
  • [Presentation] A method for constructing databases for globaldynamics of multi-parameter systems2009

    • Author(s)
      Hiroshi Kokubu
    • Organizer
      International Workshop on "What is Evolution?" - Bicentennial of Charles Darwin's Birth -
    • Place of Presentation
      COOP-INN Kyoto Conference Hall
    • Related Report
      2012 Final Research Report
  • [Presentation] A method for constructing databases for global dynamics of multi-parameter systems2009

    • Author(s)
      國府寛司
    • Organizer
      研究集会「トポロジーとコンピュータ2009」
    • Place of Presentation
      東京工業大学
    • Related Report
      2012 Final Research Report 2009 Annual Research Report
  • [Book] Capturing the global behavior of dynamical systems with Conley-Morse graphs2013

    • Author(s)
      Zin Arai, Hiroshi Kokubu, and Ippei Obayashi
    • Publisher
      Advances in Cognitive Neurodynamics (III)
    • Related Report
      2012 Final Research Report

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Published: 2009-04-01   Modified: 2019-07-29  

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