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Pulse dynamics of a 3-component reaction-diffusion system in neighborhoods of several bifurcation points

Research Project

Project/Area Number 15K04995
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Foundations of mathematics/Applied mathematics
Research InstitutionUniversity of Toyama

Principal Investigator

IKEDA HIDEO  富山大学, 理学部, 客員教授 (60115128)

Project Period (FY) 2015-04-01 – 2020-03-31
Project Status Completed (Fiscal Year 2019)
Budget Amount *help
¥4,680,000 (Direct Cost: ¥3,600,000、Indirect Cost: ¥1,080,000)
Fiscal Year 2018: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2017: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2016: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2015: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Keywords3成分反応拡散系 / 特異摂動法 / 中心多様体理論 / 縮約系 / パルスダイナミクス / スポットダイナミクス / ドリフト分岐 / Bogdanov-Takens分岐 / 縮約理論 / 進行波解 / ホップ分岐 / サドルノード分岐 / 反応拡散系 / 中心多様体縮約 / サドル-ノード分岐
Outline of Final Research Achievements

The existence and stability of localized stationary solutions in a 3-component reaction-diffusion system in R or RxR is clarified, from which we find that there exist the following three types of destabilization: (1) drift bifurcation, (2) Hopf bifurcation and (3) saddle-node bifurcation. In neighborhoods of single or double bifurcation points, reduced ODE systems are given by using center manifold reduction procedure. These coefficients of the ODE systems are determined correctly. As an application of it, the dynamics of the interaction between bifurcated solutions and heterogeneity in the media are also considered.

Academic Significance and Societal Importance of the Research Achievements

複雑な偏微分方程式で記述された解のダイナミクスを正確に縮約された中心多様体上の常微分方程式系の解の挙動で見ることが出来るので,数学を専門としない科学者や技術者に対しても非常に受け入れられ易く,学際的にも重要な意味を持ち,理学,工学,医学などの他分野への波及効果は大きいと期待している。
一番の独創的な結果は縮約された常微分方程式系が正確に与えられる点であり,特に空間非一様性の影響を明らかにする為には常微分方程式系の係数の正確な値が必要である。そのおかげで,様々なダイナミクスの分水嶺解(セパレータ)の存在も容易に見出すこが可能となった。

Report

(6 results)
  • 2019 Annual Research Report   Final Research Report ( PDF )
  • 2018 Research-status Report
  • 2017 Research-status Report
  • 2016 Research-status Report
  • 2015 Research-status Report
  • Research Products

    (36 results)

All 2020 2019 2018 2017 2016 2015 Other

All Int'l Joint Research (10 results) Journal Article (3 results) (of which Int'l Joint Research: 2 results,  Peer Reviewed: 3 results,  Acknowledgement Compliant: 1 results) Presentation (23 results) (of which Int'l Joint Research: 8 results,  Invited: 19 results)

  • [Int'l Joint Research] Leiden University(オランダ)

    • Related Report
      2019 Annual Research Report
  • [Int'l Joint Research] Queensland University of Technology(オーストラリア)

    • Related Report
      2019 Annual Research Report
  • [Int'l Joint Research] Universitat Bremen(ドイツ)

    • Related Report
      2019 Annual Research Report
  • [Int'l Joint Research] Queensland University of Technology(オーストラリア)

    • Related Report
      2018 Research-status Report
  • [Int'l Joint Research] Leiden University(オランダ)

    • Related Report
      2018 Research-status Report
  • [Int'l Joint Research] University of Bremen(ドイツ)

    • Related Report
      2018 Research-status Report
  • [Int'l Joint Research] Queensland University of Tecnology(Australia)

    • Related Report
      2017 Research-status Report
  • [Int'l Joint Research] Leiden University(Netherlands)

    • Related Report
      2017 Research-status Report
  • [Int'l Joint Research] Queensland University of Tecnology(Australia)

    • Related Report
      2016 Research-status Report
  • [Int'l Joint Research] Leiden University(Netherlands)

    • Related Report
      2016 Research-status Report
  • [Journal Article] Unfolding symmetric Bogdanov-Takens bifurcations for front dynamics in a reaction-diffusion system2019

    • Author(s)
      M.Chirilus-Bruckner, P.van Heijster, H. Ikeda and J.D.M. Rademacher
    • Journal Title

      Journal of Nonlinear Science

      Volume: 29 Issue: 6 Pages: 2911-2953

    • DOI

      10.1007/s00332-019-09563-2

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Existence and stability of singularly perturbed standing pulse solutions of a three-component FitzHugh-Nagumo system2017

    • Author(s)
      Hideo Ikeda, Yuki Akama
    • Journal Title

      Toyama Mathematical Journal

      Volume: 39 Pages: 19-85

    • Related Report
      2017 Research-status Report
    • Peer Reviewed
  • [Journal Article] Shadow system approach to a plankton model generating harmful algal bloom2017

    • Author(s)
      Hideo Ikeda, Masayasu Mimura and Tommaso Scotii
    • Journal Title

      Discrete and Continuous Dynamical Systems

      Volume: 37 Issue: 2 Pages: 829-858

    • DOI

      10.3934/dcds.2017034

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
  • [Presentation] 非一様性とフロント進行波解の相互作用ダイナミクス2020

    • Author(s)
      池田榮雄
    • Organizer
      北陸応用数理研究会2020
    • Related Report
      2019 Annual Research Report
    • Invited
  • [Presentation] 反応拡散系と特異摂動法2019

    • Author(s)
      池田榮雄
    • Organizer
      Workshop on reaction-diffusion equations and numerical analysis - In memory of Professor Yuzo Hosono -
    • Related Report
      2019 Annual Research Report
    • Invited
  • [Presentation] Bifurcation of non-trivial traveling wave solutions in a 3-component competition-diffusion system2019

    • Author(s)
      Hideo Ikeda
    • Organizer
      International Workshop on Biomathematics and Biostatistics in Taiwan
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] 反応拡散系における進行波解とそのダイナミクス2019

    • Author(s)
      池田榮雄
    • Organizer
      反応拡散系のパターンダイナミクス2
    • Related Report
      2018 Research-status Report
    • Invited
  • [Presentation] 反応拡散系における3重0固有値近傍でのフロントダイナミクス2019

    • Author(s)
      池田榮雄
    • Organizer
      非線形現象の数理解析
    • Related Report
      2018 Research-status Report
    • Invited
  • [Presentation] Complex Dynamics of Bifurcating Front Solutions in a Three-Component Reaction-Diffusion System2018

    • Author(s)
      Hideo Ikeda
    • Organizer
      The 12th AIMS Conference on Dynamical Systems, Differential Equations and Applications
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Front and pulse dynamics in a 3-component FitzHugh-Nagumo system2018

    • Author(s)
      Hideo Ikeda
    • Organizer
      Second joint Australia-Japan workshop on dynamical systems with applications in life sciences
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Front dynamics in a 3-component reaction-diffusion system2018

    • Author(s)
      池田榮雄
    • Organizer
      富山解析セミナー2018
    • Related Report
      2018 Research-status Report
  • [Presentation] 3種競争拡散系における3種共存進行波解の分岐について2018

    • Author(s)
      池田榮雄
    • Organizer
      反応拡散系の理論と応用 ~現状と未来~
    • Related Report
      2018 Research-status Report
    • Invited
  • [Presentation] Traveling pulse solutions on 3-component FitzHugh-Nagumo systems2017

    • Author(s)
      Hideo Ikeda
    • Organizer
      One-day workshop on reaction-diffusion equations at Asahikawa
    • Related Report
      2017 Research-status Report
    • Invited
  • [Presentation] 3種競争拡散系における3種共存進行波解の分岐について2017

    • Author(s)
      池田榮雄
    • Organizer
      富山解析セミナー2017
    • Related Report
      2017 Research-status Report
  • [Presentation] Complex Dynamics of Bifurcating Front Solutions in a Three-Component Reaction-Diffusion System2017

    • Author(s)
      池田 榮雄
    • Organizer
      研究集会「拡散成分と非拡散成分が共存する反応拡散系がつくるパターン」
    • Place of Presentation
      東北大学 (仙台市)
    • Related Report
      2016 Research-status Report
    • Invited
  • [Presentation] Complex Dynamics of Bifurcating Front Solutions in a Three-Component Reaction-Diffusion System2016

    • Author(s)
      Hideo Ikeda
    • Organizer
      Joint Australia-Japan workshop on dynamical systems with applications in life sciences
    • Place of Presentation
      Queensland University of Technology (Brisbane)
    • Year and Date
      2016-07-17
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Stability of standing planar spot solutions in three-component FitzHugh-Nagumo systems2016

    • Author(s)
      Hideo Ikeda
    • Organizer
      11th AIMS International Conference on Dynamical Syatems, Differential Equations and Applicatiions
    • Place of Presentation
      Hyatt Regency Orlando (Florida)
    • Year and Date
      2016-07-01
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Shadow system approach to a plankton model generating Harmuful Algal Bloom2016

    • Author(s)
      Hideo Ikeda
    • Organizer
      11th AIMS International Conference on Dynamical Syatems, Differential Equations and Applicatiions
    • Place of Presentation
      Hyatt Regency Orlando (Florida)
    • Year and Date
      2016-07-01
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] 3種競争拡散系における自明進行波解からの3種共存進行波解のドリフト分岐2016

    • Author(s)
      池田 榮雄
    • Organizer
      Workshop on reaction diffusion equations and numerical analysis
    • Place of Presentation
      京都産業大学 (京都市)
    • Related Report
      2016 Research-status Report
    • Invited
  • [Presentation] 3種競争拡散系における2種共存進行波解からの3種共存進行波解のドリフト分岐2016

    • Author(s)
      池田 榮雄
    • Organizer
      富山解析セミナー2016
    • Place of Presentation
      富山大学 (富山市)
    • Related Report
      2016 Research-status Report
  • [Presentation] 2種類の3変数系の理論的考察2015

    • Author(s)
      池田榮雄
    • Organizer
      多変数反応拡散系の数理とその周辺
    • Place of Presentation
      神戸大学海事科学部、深江キャンパス 4 号館 4207 教室
    • Year and Date
      2015-12-25
    • Related Report
      2015 Research-status Report
    • Invited
  • [Presentation] Stability analysis of standing planar spot solutions in a 3-component FitzHugh-Nagumo system2015

    • Author(s)
      池田榮雄
    • Organizer
      International Conference on Mathematical Modeling and Applications 2015`Self-Organization-Modeling and Analysis'
    • Place of Presentation
      明治大学中野キャンパス
    • Year and Date
      2015-10-26
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Existence and stability of a standing spot solution in 3-component FitzHugh-Nagumo systems2015

    • Author(s)
      池田榮雄
    • Organizer
      第14回自己組織化セミナー
    • Place of Presentation
      明治大学中野キャンパス 6階 研究セミナー室 3
    • Year and Date
      2015-10-19
    • Related Report
      2015 Research-status Report
    • Invited
  • [Presentation] 有害藻類開花モデルのshadow 近似解に関して2015

    • Author(s)
      池田榮雄
    • Organizer
      富山解析セミナー2015
    • Place of Presentation
      富山大学理学部数学科情報数理実習室B121
    • Year and Date
      2015-10-10
    • Related Report
      2015 Research-status Report
  • [Presentation] Shadow system approach to a 3-component Lotka-Volterra system with diffusion2015

    • Author(s)
      池田榮雄
    • Organizer
      第40回偏微分方程式論札幌シンポジウム
    • Place of Presentation
      北海道大学理学部 7号館講義室
    • Year and Date
      2015-08-19
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] 3種Lotka-Volterraモデルの2つの極限問題に関する定常解と安定性について2015

    • Author(s)
      池田榮雄
    • Organizer
      拡散に付随する数理科学セミナー
    • Place of Presentation
      九州大学 産学官連携本部 産学官連携イノベーションプラザ
    • Year and Date
      2015-05-30
    • Related Report
      2015 Research-status Report
    • Invited

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Published: 2015-04-16   Modified: 2021-02-19  

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