Research on the bifurcation and renormalization of dynamical systems

2014 Fiscal Year Final Research Report

• Project/Area Number: 22340033
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Global analysis
• Research Institution: Kyoto University
• Principal Investigator: SHISHIKURA Mitsuhiro 京都大学, 理学(系)研究科(研究院), 教授 (70192606)
• Co-Investigator(Kenkyū-buntansha): UEDA Tetsuo 京都大学, 大学院理学研究科, 教授 (10127053)
• Co-Investigator(Renkei-kenkyūsha): INOU Hiroyuki 京都大学, 大学院理学研究科, 講師 (00362434)
• Project Period (FY): 2010-04-01 – 2015-03-31
• Keywords: 力学系 / カオス / フラクタル / 分岐 / くりこみ

Outline of Final Research Achievements

We studied the bifurcation of chaotic dynamical systems, especially low dimensional complex dynamical systems. In order to study the bifurcation of parabolic or semi-parabolic fixed points, we established the theory of parabolic and near-parabolic renormalization, and found an invariant space of functions under these renormalizations. We introduced the notion of dynamical charts, and combining the invariant space of near-parabolic renormalization, we studied the dynamical properties and invariant sets (called hedgehogs) near irrationally invariant fixed points. We also studied the local structure and the bifurcation of semi-parabolic fixed point of two-dimensional holomorphic mappings, and obtained results on the continuity and discontinuity of the parabolic/attracting basins and unstable manifolds.

Free Research Field

数物系科学