2003 Fiscal Year Final Research Report Summary
Analysis on the structure of quasi periodic attractors for nonlinear partial differential equations
| Project/Area Number |
14540182
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Basic analysis
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| Research Institution | KUMAMOTO UNIVERSITY |
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Principal Investigator |
NAITO Koichiro Kumamoto University, Dept.Eng., Prof., 工学部, 教授 (10164104)
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| Co-Investigator(Kenkyū-buntansha) |
KADOTA Noriya Kumamoto University, Dept.Eng., Lect., 工学部, 講師 (80185884)
MISAWA Masashi Kumamoto University, Dept.Sci., A-Prof., 理学部, 助教授 (40242672)
OSHIMA Yoichi Kumamoto University, Dept.Eng., Prof., 工学部, 教授 (20040404)
SADAHIRO Taizou Kumamoto Pref. University, Dept.Adm., Lect., 総合管理学部, 講師 (00280454)
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| Project Period (FY) |
2002 – 2003
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| Keywords | nonlinear PDE / quasi periodicity / attractor / fractal dimension / Diophantine approximation / KAM theorem / self-similarity / chaos |
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| Research Abstract |
In recent years great efforts have been made to analyze complexity or chaotic behaviors in various fields. In this research we introduced recurrent dimensions of discrete dynamical systems and we have estimated the upper and lower recurrent dimensions of discrete quasi-periodic orbits to analyze complexity of quasi-periodic solutions given by various types of partial differential equaitions. We also proposed the gaps between the upper and the lower recurrent dimensions as the index parameters, which measure unpredictability levels of the orbits. We show that the gaps of recurrent dimensions of quasi-periodic orbits take positive values when the irrational frequencies are weak Liouville numbers with sufficiently large orders of goodness levels of approximation by rational numbers. These results were announced by the head investigator in the international conference NACA2003 ([1], [2]) and will appear in Discr. Conti. Dyn. Systems ([3]).
Calculating the dimensions of the attractors is to measure their level of complexity and randomness. In [5], [6], [7] the co-investigator Y. Oshima proved some related results for randomness, using probability theory. On the other hand, in [8] -[11] the co-investigator M. Misawa have shown various fundamental results on P.D.E., which will play important and essential roles for investigating chaotic behaviors of nonlinear dynamical models.
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