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On the Stability of Symmetric Periodic Orbits of the Elliptic Sitnikov Problem
SIAM Journal on Applied Dynamical Systems ( IF 1.7 ) Pub Date : 2020-05-21 , DOI: 10.1137/19m1258384 Xiuli Cen , Xuhua Cheng , Zaitang Huang , Meirong Zhang
SIAM Journal on Applied Dynamical Systems ( IF 1.7 ) Pub Date : 2020-05-21 , DOI: 10.1137/19m1258384 Xiuli Cen , Xuhua Cheng , Zaitang Huang , Meirong Zhang
SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 2, Page 1271-1290, January 2020.
Motivated by the recent works on the stability of symmetric periodic orbits of the elliptic Sitnikov problem, for time-periodic Newtonian equations with symmetries, we will study symmetric periodic solutions which emanated from nonconstant periodic solutions of autonomous equations. By using the theory of Hill's equations, we will first deduce in this paper a criterion for the linearized stability and instability of periodic solutions which are odd in time. Such a criterion is complementary to that for periodic solutions which are even in time, obtained recently by the present authors. Applying these criteria to the elliptic Sitnikov problem, we will prove in an analytical way that the odd (2,1)-periodic solutions of the elliptic Sitnikov problem are hyperbolic and therefore are Lyapunov unstable when the eccentricity is small, while the corresponding even (2,1)-periodic solutions are elliptic and linearized stable.
中文翻译：
椭圆Sitnikov问题对称周期轨道的稳定性
SIAM应用动力系统杂志，第19卷，第2期，第1271-1290页，2020年1月。
受椭圆Sitnikov问题对称周期轨道稳定性的最新研究的启发，对于具有对称性的时间周期牛顿方程，我们将研究由自治方程的非恒定周期解产生的对称周期解。通过使用希尔方程的理论，我们将首先在本文中推导出奇异时间周期解的线性化稳定性和不稳定性的判据。这样的标准是对本作者最近获得的定期解决方案的补充。将这些标准应用于椭圆Sitnikov问题，我们将以分析的方式证明椭圆Sitnikov问题的奇数（2,1）-周期解是双曲线的，因此当偏心率较小时，Lyapunov不稳定，
更新日期：2020-06-30
Motivated by the recent works on the stability of symmetric periodic orbits of the elliptic Sitnikov problem, for time-periodic Newtonian equations with symmetries, we will study symmetric periodic solutions which emanated from nonconstant periodic solutions of autonomous equations. By using the theory of Hill's equations, we will first deduce in this paper a criterion for the linearized stability and instability of periodic solutions which are odd in time. Such a criterion is complementary to that for periodic solutions which are even in time, obtained recently by the present authors. Applying these criteria to the elliptic Sitnikov problem, we will prove in an analytical way that the odd (2,1)-periodic solutions of the elliptic Sitnikov problem are hyperbolic and therefore are Lyapunov unstable when the eccentricity is small, while the corresponding even (2,1)-periodic solutions are elliptic and linearized stable.
中文翻译：
椭圆Sitnikov问题对称周期轨道的稳定性
SIAM应用动力系统杂志，第19卷，第2期，第1271-1290页，2020年1月。
受椭圆Sitnikov问题对称周期轨道稳定性的最新研究的启发，对于具有对称性的时间周期牛顿方程，我们将研究由自治方程的非恒定周期解产生的对称周期解。通过使用希尔方程的理论，我们将首先在本文中推导出奇异时间周期解的线性化稳定性和不稳定性的判据。这样的标准是对本作者最近获得的定期解决方案的补充。将这些标准应用于椭圆Sitnikov问题，我们将以分析的方式证明椭圆Sitnikov问题的奇数（2,1）-周期解是双曲线的，因此当偏心率较小时，Lyapunov不稳定，
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