Perturbation problems for degenerate integrable systems and mathematics for resonance phenomena

2016 Fiscal Year Final Research Report

• Project/Area Number: 25400108
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Kanazawa University
• Principal Investigator: Ito HIdekazu 金沢大学, 数物科学系, 教授 (90159905)
• Co-Investigator(Renkei-kenkyūsha): YAGASAKI Kazuyuki 京都大学, 情報学研究科, 教授 (40200472)
• Project Period (FY): 2013-04-01 – 2017-03-31
• Keywords: 可積分系 / 共鳴 / 超可積分系 / シンプレクティック写像 / 標準形理論

Outline of Final Research Achievements

We studied various problems concerning the global structure of solutions for integrable systems and superintegrable systems possessing larger number of integrals. The resonance among eigenvalues of linearized dynamics associated with invariant sets such as equilibria played an important role in this study. In particular, we studied an even dimensional analytic vector field near an elliptic equilibrium point and showed that there exists an analytic transformation which takes the vector field into Poincare-Dulac normal form, provided that there exist a sufficient number (associated with the resonance degree) of integrals and commuting vector fields. This gives an alternative proof of a known result under weaker assumptions in a special case. Also, we succeeded in showing that a superintegrable symplectic map can be taken analytically into Birkhoff normal form without assuming semi-simplicity of its linear part.

Free Research Field

力学系理論