Study of the complexity of invariant sets appearing in chaotic dynamical systems

2023 Fiscal Year Final Research Report

• Project/Area Number: 19K03485
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 11020:Geometry-related
• Research Institution: University of Tsukuba
• Principal Investigator: Kato Hisao 筑波大学, 数理物質系(名誉教授), 名誉教授 (70152733)
• Project Period (FY): 2019-04-01 – 2024-03-31
• Keywords: カオス力学系 / 位相次元 / エントロピー

Outline of Final Research Achievements

Takens' embedding theorem on manifolds is known to allow the reconstruction of two-sided dynamical systems from data and time series analysis. Takens' theorem is the most important in experimental science. It guarantees the possibility of reconstructing mathematical models from experiments, but its scope is limited to smooth dynamical systems on manifolds. Our theorem is a broad theorem that can be applied to one-sided dynamical systems in more complex spaces. The theorem theoretically and mathematically proves that (orbital) embedding of time series data in delay coordinates is the key to unraveling one-sided dynamical systems without any restrictions.

Free Research Field

幾何学

Academic Significance and Societal Importance of the Research Achievements

多様体上のTakensの埋込み定理が知られている。本研究で、”滑らかさを仮定しない一般的な空間と連続写像”にまで Takensの定理が拡張できることを証明した。つまり、 Takens の埋め込み定理は「topology category における再構成定理」まで拡張できる。Takens の定理は、実験データから数学モデルの再構成可能を保証する実験科学における貴重な定理であるが、多様体上の滑らかな力学系に限られていた。我々の定理は現実の世界の現象の再構築理論にとって重要である。