Geometry and statistics of homoclinic tangency in random dynamical systems

2023 Fiscal Year Final Research Report

• Project/Area Number: 19K14575
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 12020:Mathematical analysis-related
• Research Institution: Tokai University
• Principal Investigator: Nakano Yushi 東海大学, 理学部, 准教授 (50778313)
• Project Period (FY): 2019-04-01 – 2024-03-31
• Keywords: ホモクリニック接触 / ランダム力学系 / 非双曲力学系 / 転送作用素

Outline of Final Research Achievements

Various properties of dynamical systems with homoclinic tangency, which play a central role in the study of nonhyperbolic dynamical systems, were studied in their counterparts in random dynamical systems, and progress was made that was not expected at the beginning of the study. In particular, it was found that under absolute continuous noise, statistics that differ significantly from those in the noiseless case appear, such as the existence and finiteness of physical measures and the establishment of related limit theorems. We were also able to discover new phenomena related to homoclinic tangency, such as emergent phenomena and the non-existence of Lyapunov exponents.

Free Research Field

力学系・エルゴード理論

Academic Significance and Societal Importance of the Research Achievements

ホモクリニック接触を持つ力学系およびそのランダム摂動に関して、研究期間内に多くの興味深い成果が得られた点で学術的意義は高い。一方で、当初の予想に反する結果が得られたことで大きく方針転換せざるを得なくなったため、元の方向性に近い「準安定的Newhouse現象（摂動強度と吸引周期点の個数の関係）」については今後の課題となった。他方で、その中で様々な視点・手法が必要となったため、数多くの研究分野の研究者と共同研究が行われた点では、社会的な意義も十分にあったと思われる。