Studies of differential equations in dynamical systems view by means of geometric and topological methods

• Project/Area Number: 21540221
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Global analysis
• Research Institution: Kyushu University
• Principal Investigator: NII Shunsaku 九州大学, 数理(科)学研究科(研究院), 准教授 (50282421)
• Project Period (FY): 2009-04-01 – 2014-03-31
• Project Status: Completed (Fiscal Year 2013)
• Budget Amount: ¥4,550,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥1,050,000); Fiscal Year 2013: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2012: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2011: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2010: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2009: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
• Keywords: Stability Index / 分岐 / 幾何学的方法 / 位相的方法 / 相転移 / 幾何学的手法 / モース関数 / 分岐理論 / コンパクト性 / 無限次元

Research Abstract

(1)Differential equations were studied in dynamical systems view, especially, homoclinic and heteroclinic bifurcations and bifurcations and stability of traveling waves, in reaction diffusion equations in which those bifurcations naturally appear, were treated by means of geometric and topological methods. Infinite dimensional Stability Indices and infinite dimensional Maslov Indices were focused.
(2)Bifurcation analysis were carried in which the systems were assumed to be completely degenerate. Those were contrary to standard bifurcation theory in which the systems are assumed to be non-degenerate and singular points are classified.

Research Products

• [Journal Article] Bifurcations of stationary solutions with triple junctions in phase separation problems --A new view of bifurcation analysis-- (2009)
  • Author(s): Shunsaku Nii
  • Journal Title: Physica D 238 Pages: 1050-1055
  • Peer Reviewed
• [Presentation] 任意のシステムサイズに対しポテンシャルがモース関数となる隣接相互作用のgenericity (2012)
  • Author(s): 新居俊作
  • Organizer: 日本応用数理学会2012年研究部会連合発表会
  • Place of Presentation: 九州大学(招待講演)
  • Year and Date: 2012-03-09
• [Presentation] Infinite dimensional Stability Index (2010)
  • Author(s): Shunsaku Nii
  • Organizer: SIAM Conference on Nonlinear Waves and Coherent Structures
  • Place of Presentation: Sheraton Society Hill Hotel, Philadelphia, Pennsylvania, U.S.A
  • Year and Date: 2010-08-17
• [Presentation] 有限次元蔵元モデル (2010)
  • Author(s): 新居俊作
  • Organizer: 日本数学会年会
  • Place of Presentation: 慶応大学
  • Year and Date: 2010-03-27
• [Presentation] 平衡点の局所安定化による超臨界分岐の亜臨界化 (2010)
  • Author(s): 新居俊作
  • Organizer: 日本数学会年会
  • Place of Presentation: 慶応大学
  • Year and Date: 2010-03-27