AI 中文总结
本文针对平面及单连通区域中的N点涡系统，提出修正Birkhoff正规形方法，证明分层构型在无Cantor集限制下具有长时间稳定性，持续时间至少为ε^{-(n-2(N-1))}量级。
AI 中文摘要
本文研究了$\u211d^2$上及单连通区域中$N$点涡系统内分层构型的动力学与长时间稳定性，且无需KAM理论中固有的Cantor集限制。由于哈密顿扰动在通常归一化变量下缺乏标准幂级数展开，我们基于相邻变量的逐次比值发展了一种修正的Birkhoff正规形，以适应分层结构。一个关键特征是在整个正规形过程中保持所得自然指数结构。因此，对于足够大的$n$，分层构型在至少$\u03b5^{-(n-2(N-1))}$量级的时间内持续存在。
英文摘要
In this paper, we investigate the dynamics and long-time stability of hierarchical configurations in an $N$-point vortex system on $\mathbb{R}^2$ and in simply connected domains, without the Cantor-set restriction inherent in KAM theory. Since the Hamiltonian perturbation lacks a standard power-series expansion in the usual normalized variables, we develop a modified Birkhoff normal form based on successive ratios of adjacent variables, tailored to the hierarchical structure. A key feature is the preservation of the resulting natural exponent structure throughout the normal form procedure. Consequently, for $n$ sufficiently large, the hierarchical configuration persists for times of order at least $\varepsilon^{-\left(n-2(N-1)\right)}$.
