偶周期FPU链的共振标准形
Resonant normal form for even periodic FPU chains
摘要：
在本文中，我们研究周期FPU链与偶数个粒子。我们表明，在平衡点附近，任何这样的链承认\emph{resonant}伯克霍夫正常形式的顺序四，这是\emph{completely integrable} -一个重要的事实，这有助于解释费米，帕斯塔和乌拉姆的数值实验。我们分析了偶FPU链的可积逼近的矩映射。与奇数FPU链的情况不同，这些可积系统（一般）表现出双曲动力学。作为一个应用，我们证明了任何FPU链Dirichlet边界条件承认一个Birkhoff标准形的四阶，并表明KAM定理适用。
英文摘要：
In this paper we investigate periodic FPU chains with an even number of particles. We show that near the equilibrium point, any such chain admits a \emph{resonant} Birkhoff normal form of order four which is \emph{completely integrable} - an important fact which helps explain the numerical experiments of Fermi, Pasta, and Ulam. We analyze the moment map of the integrable approximation of an even FPU chain. Unlike in the case of odd FPU chains these integrable systems (generically) exhibit hyperbolic dynamics. As an application we prove that any FPU chain with Dirichlet boundary conditions admits a Birkhoff normal form up to order four and show that a KAM theorem applies.
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