Extension and Unification of Singular Perturbation Methods for ODEs Based on the Renormalization Group Method
Extension and Unification of Singular Perturbation Methods for ODEs Based on the Renormalization Group Method
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DOI：
10.1137/090745957
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发表时间：
2009-08
期刊：
影响因子：
--
通讯作者：
Hayato Chiba
中科院分区：
文献类型：
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作者：
Hayato Chiba
重整化群(RG)方法是一种用于寻找微分方程解的渐近行为的奇异摄动方法。本文考虑时间无关向量场和时间(几乎)周期向量场。证明了关于近似解的误差估计，近似不变流形的存在及其稳定性，以及从原始方程到RG方程的对称性的继承性的定理。进一步证明了RG方法统一了传统的奇异摄动方法，如平均法、多时间标度法、(超)范式理论、中心流形归约、几何奇异摄动方法和相位归约等。给出了无限级RG方程收敛的一个充要条件。
The renormalization group (RG) method is one of the singular perturbation methods which is used in searching for asymptotic behavior of solutions of differential equations. In this article, time-independent vector fields and time (almost) periodic vector fields are considered. Theorems on error estimates for approximate solutions, existence of approximate invariant manifolds and their stability, and inheritance of symmetries from those for the original equation to those for the RG equation are proved. Further, it is proved that the RG method unifies traditional singular perturbation methods, such as the averaging method, the multiple time scale method, the (hyper)normal forms theory, the center manifold reduction, the geometric singular perturbation method, and the phase reduction. A necessary and sufficient condition for the convergence of the infinite order RG equation is also investigated.