Analysis of non-hyperbolic systems through unstable periodic orbits

• Project/Area Number: 23740065
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: General mathematics (including Probability theory/Statistical mathematics)
• Research Institution: Hitotsubashi University (2013); Hokkaido University (2011-2012)
• Principal Investigator: SAIKI Yoshitaka 一橋大学, 大学院商学研究科, 准教授 (20433740)
• Project Period (FY): 2011 – 2013
• Project Status: Completed (Fiscal Year 2013)
• Budget Amount: ¥4,290,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥990,000); Fiscal Year 2013: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2012: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2011: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
• Keywords: 力学系 / カオス / 不安定周期軌道 / リアプノフベクトル / リアプノフ解析 / 多様体 / 周期軌道 / 数値計算 / 非双曲系

Research Abstract

Dynamical systems can be classified into two types hyperbolic systems and non-hyperbolic systems. Hyperbolic systems have been understood well since the Smale's work. In this study we focus on non-hyperbolic systems and obtained some results in (1) strongly non-hyperbolic case, and (2) the occurrence of non-hyperbolicity by using unstable periodic orbits. Based on these results we have constructed cycle expansion theory which works well for the Henon map of a non-hyperbolic parameter.

Research Products

• [Journal Article] 非双曲力学系の不安定周期軌道解析 : エノン写像の周期軌道展開 (2014)
  • Author(s): 斉木吉隆
  • Journal Title: 九州大学応用力学研究所研究集会報告「非線形波動現象の拡がり」 Volume: 25AO-S2 Pages: 91-100
  • Peer Reviewed
• [Journal Article] Manifold structures of Unstable Periodic Orbits and the Appearance of a Periodic Window in Chaotic Systems (2014)
  • Author(s): M. U. Kobayashi and Y. Saiki
  • Journal Title: Physical Review Volume: E 89
  • Peer Reviewed
• [Journal Article] Manifold structures of Unstable Periodic Orbits and the Appearance of a Periodic Window in Chaotic Systems (2014)
  • Author(s): Miki U Kobayashi, Yoshitaka Saiki
  • Journal Title: Physical Review E Volume: Vol.89
  • Peer Reviewed
• [Journal Article] 非双曲力学系の不安定周期軌道解析：エノン写像の周期軌道展開 (2014)
  • Author(s): 斉木吉隆
  • Journal Title: 九州大学応用力学研究所 研究集会報告「非線形波動現象の拡がり」 Volume: 25AO-S2 Pages: 91-100
  • Peer Reviewed
• [Journal Article] Time Averaged Properties Along Unstable Periodic Orbits of the Kuramoto-Sivashinsky Equation, Nankai Series in Pure (2013)
  • Author(s): Y. Saiki and M. Yamada
  • Journal Title: Applied Mathematics and Theoretical Physics, World Scientific Volume: Volume 10 Pages: 145-154
  • Peer Reviewed
• [Journal Article] Time Averaged Properties Along Unstable Periodic Orbits of the Kuramoto-Sivashinsky Equation (2013)
  • Author(s): Yoshitaka Saiki and Michio Yamada
  • Journal Title: Nankai Series in Pure, Applied Mathematics and Theoretical Physics Volume: 10 Pages: 145-154
  • Peer Reviewed
• [Journal Article] Recognition of transition patterns in a business cycle model by unstable periodic orbits (2011)
  • Author(s): Y. Saiki and K. Ishiyama
  • Journal Title: International Journal of Bifurcation and Chaos Volume: 21 Pages: 1-21
  • Peer Reviewed
• [Journal Article] Economic intermittency in a two-country model of business cycles coupled by investment (2011)
  • Author(s): Y. Saiki, A. Chian and H. Yoshida
  • Journal Title: Chaos, Solitons and Fractals Volume: 44 Pages: 418-428
  • Peer Reviewed
• [Presentation] Unstable periodic orbits in intermittent dynamics (2014)
  • Author(s): Y. Saiki
  • Organizer: Workshop on Lagrangian Coherent Structures and Dynamical Systems
  • Place of Presentation: Hokkaido University,北海道
  • Year and Date: 2014-03-07
• [Presentation] Characterization of tangencies between stable and unstable manifolds through unstable periodic orbits (2013)
  • Author(s): Y. Saiki and M. U. Kobayashi
  • Organizer: The 3rd International Symposium on Innovative Mathematical Modelling
  • Place of Presentation: The University ofTokyo,東京都
  • Year and Date: 2013-11-15
• [Presentation] 非双曲力学系の不安定周期軌道解析 (2013)
  • Author(s): 斉木吉隆
  • Organizer: 九州大学応用力学研究所共同利用研究集会「非線形波動研究の拡がり」
  • Place of Presentation: 九州大学,福岡県
  • Year and Date: 2013-11-02
• [Presentation] 非双曲力学系の解析 (2013)
  • Author(s): 斉木吉隆
  • Organizer: Hayama meeting
  • Place of Presentation: IPC生産性国際交流センター,神奈川県
  • Year and Date: 2013-09-01
• [Presentation] 離散時間カルドアモデルのトーラスとその制御 (2012)
  • Author(s): 江刺邦彦,斉木吉隆
  • Organizer: 日本地域学会
  • Place of Presentation: 立正大学,東京都
  • Year and Date: 2012-10-07
• [Presentation] Numerical study of unstable periodic orbits in some chaotic systems (2012)
  • Author(s): Y. Saiki
  • Organizer: International Conference : 100 years after Poincare
  • Place of Presentation: Spain
  • Year and Date: 2012-09-09
• [Presentation] Covatiant Lyapunov vectors and unstable periodic orbits in the '63 Lorenz system (2012)
  • Author(s): Y. Saiki
  • Organizer: Weather-Chaos Seminar
  • Place of Presentation: University of Maryland, U.S.A
  • Year and Date: 2012-08-27
• [Presentation] カオス的景気循環モデルの不安定周期軌道解析 (2012)
  • Author(s): 斉木吉隆
  • Organizer: 数理科学セミナー
  • Place of Presentation: 一橋大学,東京都
  • Year and Date: 2012-06-27
• [Presentation] カオス力学系の共変リアプノフ解析 (2012)
  • Author(s): 斉木吉隆
  • Organizer: 研究会「流体乱流の力学と統計-複雑系化学からのアプローチ」
  • Place of Presentation: しこつ湖鶴雅リゾートスパ水の謌,北海道
  • Year and Date: 2012-03-06
• [Presentation] Numerical Study of Unstable Periodic Orbits in the Kuramoto-Sivashinsly system (2011)
  • Author(s): Y. Saiki
  • Organizer: Emerging topics on Differential Equations and their Applications
  • Place of Presentation: Nankai University, China
  • Year and Date: 2011-12-09
• [Presentation] Relations between Statistical Values along Unstable Periodic Orbits in Differential Equation Systems (2011)
  • Author(s): Y. Saiki and M. Yamada
  • Organizer: IUTAM Chaos
  • Place of Presentation: Kyoto University,京都府
  • Year and Date: 2011-12-01
• [Presentation] Recognition of chaotic saddles and classification of unstable periodic orbits of the Kuramoto-Sivashinsky system (2011)
  • Author(s): Y.Saiki, M. Yamada, A. C.-L Chian, M. U. Kobayashi
  • Organizer: Dynamics Days Europe
  • Place of Presentation: Oldenburg University, Germany
  • Year and Date: 2011-09-15
• [Presentation] Reconstruction of chaotic saddles and classification of unstable periodic orbits of the Kuramoto-Sivashinsky system (2011)
  • Author(s): Yoshitaka Saiki, Michio Yamada, Abraham C.-L. Chian, Miki U. Kobayashi
  • Organizer: Dynamics Days Europe 2011
  • Place of Presentation: Oldenburg University (Germany)
• [Presentation] Numerical Study of Unstable Periodic Orbits in the Kuramoto-Sivashinsky system (2011)
  • Author(s): Yoshitaka Saiki
  • Organizer: Emerging Topics on Differential Equations and their Applications(招待講演)
  • Place of Presentation: Nankai University (China)
• [Presentation] Relations between Statistical Values along Unstable Periodic Orbits in Differential Equation Systems (2011)
  • Author(s): Yoshitaka Saiki., Michio Yamada
  • Organizer: IUTAM Choas, Kyoto University
  • Place of Presentation: Kyoto University (京都市)
• [Presentation] Unstable periodic orbits in intermittent dynamics
  • Author(s): Yoshitaka Saiki
  • Organizer: Workshop on Lagrangian Coherent Structures and Dynamical Systems
  • Place of Presentation: Hokkaido University （北海道）
  • Invited
• [Presentation] 非双曲力学系の不安定周期軌道解析
  • Author(s): 斉木吉隆
  • Organizer: 九州大学応用力学研究所 共同利用研究集会 「非線形波動研究の拡がり」
  • Place of Presentation: 九州大学 （福岡）
  • Invited
• [Presentation] Characterization of tangencies between stable and unstable manifolds through unstable periodic orbits
  • Author(s): Yoshitaka Saiki and Miki U Kobayashi
  • Organizer: The 3rd International Symposium on Innovative Mathematical Modelling
  • Place of Presentation: The University of Tokyo （東京）
• [Presentation] Covariant Lyapunov vectors and unstable periodic orbits in the ‘63 Lorenz system
  • Author(s): Yoshitaka Saiki
  • Organizer: Weather-Chaos Seminar
  • Place of Presentation: University of Maryland （USA）
  • Invited
• [Presentation] Numerical study of unstable periodic orbits in some chaotic systems
  • Author(s): Yoshitaka Saiki
  • Organizer: International Conference: 100 years after Poincare
  • Place of Presentation: Hotel NH Gijon （Spain）
• [Presentation] カオス的景気循環モデルの不安定周期軌道解析
  • Author(s): 斉木吉隆
  • Organizer: 数理科学セミナー
  • Place of Presentation: 一橋大学 （東京都国立市）
  • Invited
• [Presentation] 離散時間カルドアモデルのトーラスとその制御
  • Author(s): 江刺邦彦、斉木吉隆
  • Organizer: 日本地域学会
  • Place of Presentation: 立正大学 （東京都品川区）
• [Remarks] 研究代表者ホームページ
  • URL: http://www.cm.hit-u.ac.jp/~saiki