标题： 平面n体问题中正n边形椭圆相对平衡的谱不稳定性 显示英文标题

标题： Spectral instability of the regular n-gon elliptic relative equilibrium in the planar n-body problem

摘要： 正$n$-边形椭圆相对平衡（ERE）是由正$n$-边形中心配置生成的开普勒同宿解，其线性稳定性取决于偏心率$\mathfrak{e}\in[0,1)$。 虽然 Moeckel\cite{Moe1}在$\mathfrak{e}=0$对所有$n\geq3$确立了该解的谱不稳定性，但尚不清楚对于$\mathfrak{e} \in (0,1)$不稳定性是否仍然存在。 本文解决了这个问题：我们证明了正$n$-边形 ERE 对所有$n\geq 3$和$\mathfrak{e} \in [0,1)$都是谱不稳定性的。 此外，我们引入了与拉格朗日解相关的$\beta$-系统，并开发了一种估计方法，通过仅在有限数量的点上测试$\beta$-系统的双曲性，即可获得广泛的双曲区域。 作为推论，对于$n=3,4,5$，我们统一证明了不稳定性对于所有$\mathfrak{e} \in [0,1)$都是双曲性的（因此更强）。 显示英文摘要

摘要： The regular $n$-gon elliptic relative equilibrium (ERE) is a Kepler homographic solution generated by the regular $n$-gon central configuration, and its linear stability depends on the eccentricity $\mathfrak{e}\in[0,1)$. While Moeckel \cite{Moe1} established the spectral instability for this solution at $\mathfrak{e}=0$ for all $n\geq3$, it remained unknown whether instability persists for $\mathfrak{e} \in (0,1)$. This paper resolves this problem: we prove that the regular $n$-gon ERE is spectral instability for all $n\geq 3$ and $\mathfrak{e} \in [0,1)$. Furthermore, we introduce the $\beta$-system which related the Lagrange solution, and we developed an estimation method that, by testing the hyperbolicity of the $\beta$-system at a finite number of points alone, allows us to obtain extensive hyperbolic regions. As a corollary, for $n=3,4,5$, we uniformly demonstrate that the instability is hyperbolic (and hence stronger) for all $\mathfrak{e} \in [0,1)$.