2004 Fiscal Year Final Research Report Summary
New Development of Singular-Perturbation Method with Renormalization Group
| Project/Area Number |
13640402
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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 |
物理学一般
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| Research Institution | Nagoya University |
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Principal Investigator |
NOZAKI Kazuhiro Nagoya University, Graduate School of Science, professor, 大学院・理学研究科, 教授 (00115619)
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| Co-Investigator(Kenkyū-buntansha) |
KONISI Teturo Nagoya University, Graduate School of Science, associate professor, 大学院・理学研究科, 助教授 (30211238)
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| Project Period (FY) |
2001 – 2004
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| Keywords | Singular perturbation method / renormalization group / symplectic map / Lie symmetry / phase equation / similarity-type solution |
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| Research Abstract |
The long-time asymptotic behavior of a nonlinear dynamical system has been studied by various singular perturbation methods, such as the multi-time method, the normal form theory and the reductive perturbation method. Those various perturbation methods have been unified by the recent development of the perturbative renormalization group method (the RG method), where the long-time asymptotic behavior of a system is described by the renormalization group equation. In this work, the RG method is developed as follows.
1.A symplecticity-preserving RG method is formulated and is applied to the Poicare-Birkhoff bifurcation of a two-dimensional symplectic map. We obtain analytical expressions to the resonant island structure.
2.A proto-RG approach of the RG method is proposed to avoid the step of obtaining explicit secular solutions in the RG method. Various phase equations are systematically derived as RG equations from a general reaction-diffusion equation.
3.The RG method with the Lie symmetry is extended to apply to interesting physical systems such as a gas sphere under gravity and adiavatic perfect gas, which have only trivial Lie symmetries.
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