2025/10/30 by Ou, Yuwei, Wang, Yunying
#34L15 #37J25 #37J46 #70F10 #Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.2510.26211
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 \mathfrake∈[0,1). While Moeckel \citeMoe1 established the spectral instability for this solution at \mathfrake=0 for all n≥3, it remained unknown whether instability persists for \mathfrake ∈ (0,1). This paper resolves this problem: we prove that the regular n-gon ERE is spectral instability for all n≥ 3 and \mathfrake ∈ [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 \mathfrake ∈ [0,1).