SIAM Journal on Applied Dynamical Systems, Volume 20, Issue 2, Page 941-952, January 2021.
In this paper, we aim at proving a conjecture proposed in [X. Cen, X. Cheng, Z. Huang, and M. Zhang, SIAM J. Appl. Dyn. Syst., 19 (2020), pp. 1271--1290] on the stability of symmetric periodic solutions of the elliptic Sitnikov problem $(S_e)$. By using the stability criteria there, it is shown that odd $4n\pi$-periodic solutions are hyperbolic and therefore are unstable when the eccentricity $e$ is small; meanwhile, even $(8n-4)\pi$- and even $8n\pi$-periodic solutions are, respectively, elliptic and hyperbolic when the eccentricity $e$ is small. Here $n$ is an arbitrary positive integer, and the range of the eccentricities is dependent on $n$.

中文翻译：

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椭圆型Sitnikov问题对称周期解的稳定性猜想的证明

SIAM应用动力系统杂志，第20卷，第2期，第941-952页，2021
年1月。Cen，X.Cheng，Z.Huang，and M.Zhang，SIAM J.Appl。达因 Syst。，19（2020），pp。1271--1290]讨论了椭圆Sitnikov问题$（S_e）$的对称周期解的稳定性。通过使用那里的稳定性标准，表明奇数$ 4n \ pi $-周期解是双曲的，因此当偏心率$ e $小时是不稳定的；同时，当偏心率$ e $小时，甚至$（8n-4）\ pi $-和$ 8n \ pi $周期解也分别是椭圆形和双曲线形。这里$ n $是一个任意的正整数，并且离心率的范围取决于$ n $。