力学系：理論と応用の新展開

• Project/Area Number: 18H03671
• Research Category: Grant-in-Aid for Scientific Research (A)
• Allocation Type: Single-year Grants
• Section: 一般
• Review Section: Medium-sized Section 12:Analysis, applied mathematics, and related fields
• Research Institution: Kyoto University
• Principal Investigator: 國府 寛司 京都大学, 理学研究科, 教授 (50202057)
• Co-Investigator(Kenkyū-buntansha): 角 大輝 京都大学, 人間・環境学研究科, 教授 (40313324) ; 荒井 迅 中部大学, 創発学術院, 教授 (80362432)
• Project Period (FY): 2018-04-01 – 2023-03-31
• Project Status: Granted (Fiscal Year 2022)
• Budget Amount: ¥43,810,000 (Direct Cost: ¥33,700,000、Indirect Cost: ¥10,110,000); Fiscal Year 2022: ¥8,970,000 (Direct Cost: ¥6,900,000、Indirect Cost: ¥2,070,000); Fiscal Year 2021: ¥8,710,000 (Direct Cost: ¥6,700,000、Indirect Cost: ¥2,010,000); Fiscal Year 2020: ¥8,320,000 (Direct Cost: ¥6,400,000、Indirect Cost: ¥1,920,000); Fiscal Year 2019: ¥7,930,000 (Direct Cost: ¥6,100,000、Indirect Cost: ¥1,830,000); Fiscal Year 2018: ¥9,880,000 (Direct Cost: ¥7,600,000、Indirect Cost: ¥2,280,000)
• Keywords: 力学系 / 分岐 / 大域的構造 / ランダム / 時系列

Outline of Annual Research Achievements

研究計画に従い，2019年度から2021年度（2019年度の繰越分と事故繰越分）に行った研究は以下のとおりである：
課題(LDF=Large Degrees of Freedom systems)では、國府は海外共同研究者 K. Mischaikow らと遺伝子制御ネットワークの1つの数理モデル化である switching system と呼ばれる区分的線型の結合ODE系のダイナミクスの研究を進めた。また分担者の荒井は、高次元力学系の構造を位相力学系と複素力学系の双方の視点から解析する研究を進め、特にエノン写像のパラメータ空間の位相的な構造についての新しい結果を得た。
課題(RSV=Random and Set-Valued dynamical systems)では分担者の角はランダム複素力学系の研究を行った。特に、独立同分布とは限らない、マルコフ的ランダム力学系の基礎理論を構築し、その安定性などについて考察し、論文にまとめて発表した。
課題(HPS=Hybrid and Piecewise Smooth systems)では、國府は岡本耕太・青井伸也らと共同で歩行・走行遷移に関係するハイブリッド力学系の分岐の解析を行った。
課題(DTD=Dynamical Time-series and Data)では、國府は大学院生の原誠人と共同で、リザバー計算と呼ばれる機械学習手法を用いて、力学系の時系列データからリザバー計算により元の力学系の情報が復元される数学的機構についての数値的研究を行った。特に力学系を学習したリザバー計算システムが、学習対象の力学系の情報を超えて、学習データに含まれない相空間構造や分岐の情報までも復元することを、ロジスティック写像と呼ばれるカオス力学系の場合に精密に研究した。

Current Status of Research Progress

3: Progress in research has been slightly delayed.

Reason

2020年初に始まったコロナ禍で、2019年度末に予定していた研究打合せと成果発表のための海外渡航や外国人共同研究者の招へいができなくなり、研究費の繰越が不可避となった。また、それ以後２年以上にわたり現在も続くコロナ感染拡大の影響で、特に2020年度は研究計画の大幅な練り直しを余儀なくされ、その影響は今に及んでいる。
そのような困難下でも、海外研究者との研究連絡や国際研究集会での成果発表はオンラインでの実施が普及しており、また研究の遅れを取り戻すために特定研究員などの雇用を行なっている。その結果、当初の研究計画は２年の遅れを経て概ね実施できたと考えている。

Strategy for Future Research Activity

コロナ禍の今後の状況はまだ予断を許さないが、それぞれの課題について、研究分担者とも緊密に協力しつつ、研究計画に沿って着実に研究を進める。

Research Products

• [Int'l Joint Research] Rutgers University/Montana State University(米国)
• [Int'l Joint Research] Imperial College London(英国)
• [Journal Article] Equilibria and their Stability in Networks with Steep Sigmoidal Nonlinearities (2021)
  • Author(s): Duncan William、Gedeon Tomas、Kokubu Hiroshi、Mischaikow Konstantin、Oka Hiroe
  • Journal Title: SIAM J. Applied Dynamical Systems Volume: 20 Pages: 2108-2141
  • DOI: 10.1137/21m1398331
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] Fractal mechanism of basin of attraction in passive dynamic walking (2020)
  • Author(s): Okamoto Kota、Aoi Shinya、Obayashi Ippei、Kokubu Hiroshi、Senda Kei、Tsuchiya Kazuo
  • Journal Title: Bioinspiration & Biomimetics Volume: 15 Pages: 055002-055002
  • DOI: 10.1088/1748-3190/ab9283
  • NAID: 120006888011
  • Peer Reviewed / Open Access
• [Journal Article] Visualization of the Dynamics Effect: Projection of on-the-Fly Trajectories to the Subspace Spanned by the Static Reaction Path Network (2020)
  • Author(s): Tsutsumi Takuro, Ono Yuriko, Arai Zin, Taketsugu Tetsuya
  • Journal Title: Journal of Chemical Theory and Computation Volume: 16 Pages: 4029-4037
  • DOI: 10.1021/acs.jctc.0c00018
  • NAID: 120007044513
  • Peer Reviewed
• [Journal Article] Hereditarily non uniformly perfect non-autonomous Julia sets (2020)
  • Author(s): Comerford Mark、Stankewitz Rich、Sumi Hiroki
  • Journal Title: Discrete and Continuous Dynamical Systems Volume: 40 Pages: 33-33
  • DOI: 10.3934/dcds.2020002
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] The Hausdorff dimension function of the family of conformal iterated function systems of generalized complex continued fractions (2020)
  • Author(s): Inui Kanji、Okada Hikaru、Sumi Hiroki
  • Journal Title: Discrete and Continuous Dynamical Systems - A Volume: 40 Pages: 753-766
  • DOI: 10.3934/dcds.2020060
  • Peer Reviewed
• [Journal Article] Hausdorff measures and packing measures of limit sets of CIFSs of generalized complex continued fractions (2020)
  • Author(s): Inui Kanji、Sumi Hiroki
  • Journal Title: Journal of Difference Equations and Applications Volume: 26 Pages: 104-121
  • DOI: 10.1080/10236198.2019.1709063
  • Peer Reviewed
• [Journal Article] Explicit transversality conditions and local bifurcation diagrams for Bogdanov-Takens bifurcation on center manifolds (2019)
  • Author(s): Yang Li, Hiroshi Kokubu, and Kazuyuki Aihara
  • Journal Title: Physica D: Nonlinear Phenomena Volume: 391 Pages: 52-65
  • DOI: 10.1016/j.physd.2018.11.005
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] The onset of transient turbulence in minimal plane Couette ow (2019)
  • Author(s): Julius Rhoan Lustro, Genta Kawahara, Lennaert van Veen, Masaki Shimizu, Hiroshi Kokubu
  • Journal Title: J. Fluid Mech. Rapids Volume: 862 Pages: 1-15
  • DOI: 10.1017/jfm.2018.971
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] Topological computation analysis of meteorological time-series data. (2019)
  • Author(s): Morita, H., M. Inatsu, and H. Kokubu
  • Journal Title: SIAM Journal on Applied Dynamical Systems Volume: 18 Pages: 1200-1212
  • DOI: 10.1137/18m1184746
  • Peer Reviewed / Open Access
• [Journal Article] Non-i.i.d. random holomorphic dynamical systems and the probability of tending to infinity (2019)
  • Author(s): Sumi Hiroki、Watanabe Takayuki
  • Journal Title: Nonlinearity Volume: 32 Pages: 3742-3771
  • DOI: 10.1088/1361-6544/ab231e
  • Peer Reviewed
• [Journal Article] Hereditarily non-uniformly perfect sets (2019)
  • Author(s): R. Stankewitz, T. Sugawa and H. Sumi
  • Journal Title: Discrete Contin. Dyn. Syst. Ser. S Volume: 12 Pages: 2391-2402
  • DOI: 10.3934/dcdss.2019150
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] On Parameter Loci of the Henon Family (2018)
  • Author(s): Arai Zin、Ishii Yutaka
  • Journal Title: Communications in Mathematical Physics Volume: 361 Pages: 343-414
  • DOI: 10.1007/s00220-018-3174-0
  • NAID: 120006940730
  • Peer Reviewed / Open Access
• [Journal Article] Boundary of the horseshoe locus for the Henon family (2018)
  • Author(s): Z. Arai, Y. Ishii, H. Takahasi
  • Journal Title: SIAM Journal on Applied Dynamical Systems Volume: 17 Pages: 2234-2248
  • DOI: 10.1137/18m1174684
  • Peer Reviewed / Open Access
• [Journal Article] A visualization of the intrinsic reaction coordinate and global reaction route map by classical multidimensional scaling (2018)
  • Author(s): Takuro Tsutsumi, Yuriko Ono, Zin Arai, Tetsuya Taketsugu
  • Journal Title: J. Chem. Theory Comput. Volume: 14 Pages: 4263-4270
  • DOI: 10.1021/acs.jctc.8b00176
  • Peer Reviewed
• [Presentation] Learning dynamics of the logistic maps by Reservoir Computing (2021)
  • Author(s): Hiroshi Kokubu
  • Organizer: International Conference: Differential Equations for Data Science 2021
  • Int'l Joint Research / Invited
• [Presentation] 力学系のリザバー計算 (2021)
  • Author(s): 國府寛司
  • Organizer: 東京理科大学 第２回数理・AIセミナー
  • Invited
• [Presentation] 力学系のリザバー計算：ロジスティック写像の場合 (2021)
  • Author(s): 國府寛司
  • Organizer: 日本数学会秋季総合分科会
• [Presentation] 力学系のリザバー計算：ある数学的枠組み (2021)
  • Author(s): 國府寛司
  • Organizer: 日本数学会秋季総合分科会
• [Presentation] Period doubling bifurcations from complex and algebraic point of view (2020)
  • Author(s): Z. Arai
  • Organizer: Rutgers University Dynamical System Seminar
• [Presentation] Non-uniformly perfect Julia sets of non-autonomous dynamics of unbounded sequences of polynomials (2020)
  • Author(s): 角大輝
  • Organizer: 2019年度冬の力学系研究集会(2020年1月10日－13日）、日本大学軽井沢研修所
• [Presentation] Period doubling bifurcations from complex and algebraic points of view (2020)
  • Author(s): Zin Arai
  • Organizer: Dynamical Systems Seminar, Rutgers University, USA
  • Invited
• [Presentation] Morse decomposition of dynamics from time-series data (2019)
  • Author(s): Hiroshi Kokubu
  • Organizer: Workshop on Data Analysis and Machine Learning in Dynamical Systems
  • Int'l Joint Research / Invited
• [Presentation] Classification of generic random holomorphic dynamical systems associated with analytic families of rational maps (2019)
  • Author(s): Hiroki Sumi
  • Organizer: Thermodynamic Formalism – Applications to Geometry, Number Theory, and Stochastics
  • Int'l Joint Research / Invited
• [Presentation] Classification of generic random holomorphic dynamical systems associated with analytic families of rational maps (2019)
  • Author(s): Hiroki Sumi
  • Organizer: The 2019 Fall Program of Low-Dimensional Dynamics, Shanghai Center for Mathematical Sciences
  • Int'l Joint Research / Invited
• [Presentation] Classification of generic random holomorphic dynamical systems associated with analytic families of rational maps (2019)
  • Author(s): 角大輝
  • Organizer: 数理解析研究所研究集会「ランダム力学系理論とフラクタル幾何学の研究」
• [Presentation] Classification of generic random holomorphic dynamical systems associated with analytic families of rational maps (2019)
  • Author(s): 角大輝
  • Organizer: 第９回岐阜数理科学研究会
  • Invited
• [Presentation] Classification of generic random holomorphic dynamical systems associated with analytic families of rational maps (2019)
  • Author(s): 角大輝
  • Organizer: 研究集会「エルゴード理論とその周辺」
• [Presentation] Classification of generic random holomorphic dynamical systems associated with analytic families of rational maps (2019)
  • Author(s): 角大輝
  • Organizer: 数理解析研究所研究集会「複素力学系の分岐と安定性の研究」
  • Invited
• [Presentation] On the gradient structure of chemical reactions (2019)
  • Author(s): 荒井迅
  • Organizer: RIMS 研究集会: 力学系–新たな理論と応用に向けて–
• [Presentation] On period doubling bifurcations (2019)
  • Author(s): 荒井迅
  • Organizer: 第 9 回岐阜数理科学研究会
  • Invited
• [Presentation] Classification of generic random holomorphic dynamical systems associated with analytic families of rational maps (2019)
  • Author(s): Hiroki Sumi
  • Organizer: Workshop on Random Dynamical Systems and Anomalous Dynamics, Imperial College London, ロンドン、イギリス
  • Int'l Joint Research / Invited
• [Presentation] Classification of generic random holomorphic dynamical systems associated with analytic families of rational maps (2019)
  • Author(s): 角大輝
  • Organizer: ２０１８年度冬の力学系研究集会、 日本大学軽井沢研修所
• [Presentation] On the monodromy and period doubling bifurcations of the He'non map (2019)
  • Author(s): Zin Arai
  • Organizer: Real and Complex Dynamics of He'non's Map, 慶應大学, 2019年3月
  • Int'l Joint Research / Invited
• [Presentation] Morse decomposition of dynamics from time-series data, (2019)
  • Author(s): Hiroshi Kokubu
  • Organizer: Workshop on Data Analysis and Machine Learning in Dynamical Systems, Faculty of Sciences, University of Tokyo
  • Int'l Joint Research / Invited
• [Presentation] Morse decomposition of dynamics from time-series data and its application (2019)
  • Author(s): Hiroshi Kokubu
  • Organizer: EASIAM Wuhan University, Wuhan, China June 13-16, 2019
  • Int'l Joint Research / Invited
• [Presentation] Classification of generic random holomorphic dynamical systems associated with analytic families of rational maps (2019)
  • Author(s): Hiroki Sumi
  • Organizer: Thermodynamic Formalism; Applications to Geometry, Number Theory, and Stochastics (2019年7月8日ー12日）, Institut Mittag-Leffler, スウェーデン
  • Int'l Joint Research / Invited
• [Presentation] Classification of generic random holomorphic dynamical systems associated with analytic families of rational maps (2019)
  • Author(s): Hiroki Sumi
  • Organizer: The 2019 Fall Program of Low-Dimensional Dynamics, Shanghai Center for Mathematical Sciences （2019年9月23日ー27日）, Fudan University, 上海、中国
  • Int'l Joint Research / Invited
• [Presentation] Classification of generic random holomorphic dynamical systems associated with analytic families of rational maps (2019)
  • Author(s): Hiroki Sumi
  • Organizer: 数理解析研究所研究集会「Research on the Theory of Random Dynamical Systems and Fractal Geometry」（2019年8月29日ー9月2日）、京都大学
  • Int'l Joint Research
• [Presentation] Classification of generic random holomorphic dynamical systems associated with analytic families of rational maps (2019)
  • Author(s): 角大輝
  • Organizer: 第９回岐阜数理科学研究会(2020年9月9日--11日）、飛騨高山まちの博物館研修室、岐阜県高山市
  • Invited
• [Presentation] Classification of generic random holomorphic dynamical systems associated with analytic families of rational maps (2019)
  • Author(s): 角大輝
  • Organizer: 研究集会「エルゴード理論とその周辺」（2020年11月20日ー23日）、 新潟県長岡市まちなかキャンパス
• [Presentation] Random holomorphic dynamical systems associated with analytic families of holomorphic maps (2019)
  • Author(s): 角大輝
  • Organizer: 数理解析研究所研究集会「複素力学系の分岐と安定性の研究」（2020年12月9日--13日）、数理解析研究所
  • Int'l Joint Research / Invited
• [Presentation] On the gradient structure of chemical reactions (2019)
  • Author(s): 荒井迅
  • Organizer: 力学系―新たな理論と応用に向けて―, 数理解析研究所
• [Presentation] On period doubling bifurcations (2019)
  • Author(s): 荒井迅
  • Organizer: 第9回岐阜数理科学研究会, 飛騨高山まちの博物館
  • Invited
• [Presentation] 計算トポロジーと位相的データ解析 (2019)
  • Author(s): 荒井迅
  • Organizer: 東京工業大学情報理工学院寺子屋セミナー, 東京工業大学
  • Invited
• [Presentation] Computer-assisted Approach to Global Dynamics of Multi-parameter Systems (2018)
  • Author(s): Hiroshi Kokubu
  • Organizer: KSIAM Spring Conference KAIST, Daejeon, South Korea
  • Int'l Joint Research / Invited
• [Presentation] Morse decomposition of the switching system and its smooth perturbations (2018)
  • Author(s): Hiroshi Kokubu
  • Organizer: International Conference on Algebraic Topology in Data and Dyamics, Montana State University, Bozeman, USA
  • Int'l Joint Research / Invited
• [Presentation] Classification of generic random holomorphic dynamical systems associated with analytic families of rational maps (2018)
  • Author(s): Hiroki Sumi
  • Organizer: The 12th AIMS Conference on Dynamical Systems, Differential Equations and Applications, 台北
  • Int'l Joint Research
• [Presentation] Classification of generic random holomorphic dynamical systems associated with analytic families of rational maps (2018)
  • Author(s): 角大輝
  • Organizer: 研究集会「エルゴード理論とその周辺」、大阪大学
  • Invited
• [Presentation] Classification of generic random holomorphic dynamical systems associated with analytic families of rational maps (2018)
  • Author(s): 角大輝
  • Organizer: 複素力学系研究とその発展 、京都大学数理解析研究所
  • Int'l Joint Research
• [Presentation] The Conley index for the real and complex He'non maps (2018)
  • Author(s): Zin Arai
  • Organizer: International Conference on Algebraic Topology in Data and Dyamics, Montana State University, Bozeman, USA
  • Int'l Joint Research
• [Presentation] 力学系の大域的な分岐とその計算について (2018)
  • Author(s): 荒井迅
  • Organizer: 岩手数理科学セミナー， 岩手大学
  • Invited
• [Presentation] 化学反応経路網の幾何学的理解に向けて (2018)
  • Author(s): 荒井迅
  • Organizer: 日本応用数理学会2018年度年会, 名古屋大学
• [Funded Workshop] 数理解析研究所研究集会「Research on the Theory of Random Dynamical Systems and Fractal Geometry」 (2019)