Spectral instability of the regular n-gon elliptic relative equilibrium in the planar n-body problem
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| Format: | Preprint |
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2025
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| _version_ | 1866914124115476480 |
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| author | Ou, Yuwei Wang, Yunying |
| author_facet | Ou, Yuwei Wang, Yunying |
| contents | 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 $β$-system which related the Lagrange solution, and we developed an estimation method that, by testing the hyperbolicity of the $β$-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 $\mathfrak{e} \in [0,1)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_26211 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Spectral instability of the regular n-gon elliptic relative equilibrium in the planar n-body problem Ou, Yuwei Wang, Yunying Dynamical Systems Classical Analysis and ODEs 70F10, 37J25, 37J46, 34L15 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 $β$-system which related the Lagrange solution, and we developed an estimation method that, by testing the hyperbolicity of the $β$-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 $\mathfrak{e} \in [0,1)$. |
| title | Spectral instability of the regular n-gon elliptic relative equilibrium in the planar n-body problem |
| topic | Dynamical Systems Classical Analysis and ODEs 70F10, 37J25, 37J46, 34L15 |
| url | https://arxiv.org/abs/2510.26211 |