不等质量等边限制性四体问题中的非线性稳定性、共振与奇异约化
Nonlinear Stability, Resonances, and Singular Reduction in the Unequal-Mass Equilateral Restricted Four-Body Problem
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中文总结 AI 辅助
该研究针对不等质量等边限制性四体问题，结合高阶Birkhoff规范形等方法分析椭圆平衡点的非线性稳定性，拓展了对称质量构型外的稳定性分析并给出共振动力学的几何描述。
中文摘要 AI 辅助
我们研究了具有不等主质量的平面等边限制性四体问题中椭圆平衡点$L_3$、$L_5$和$L_6$的非线性稳定性。通过结合高阶Birkhoff规范形、数值延拓以及共振规范形的奇异约化，我们在完整的双参数质量平面上对非线性稳定性进行了系统分类。在非共振区域，当四次非退化条件成立时应用Arnold定理，而六阶归一化则解决了退化情况。利用Alfriend、Markeev和Meyer的定理分析了2:$-1$和3:$-1$共振，包括对3:$-1$共振动力学的分类。奇异约化提供了对应的约化轨道空间，并揭示了与周期轨道族出现和消失相关的鞍点-中心分岔。这些结果将先前的稳定性分析扩展到了对称质量构型之外，并为不等质量等边限制性四体问题中的共振动力学提供了几何描述。
英文摘要
We study the nonlinear stability of the elliptic equilibrium points $L_3$, $L_5$, and $L_6$ in the planar equilateral restricted four-body problem with unequal primary masses. We provide a systematic classification of nonlinear stability over the full two-parameter mass plane by combining high-order Birkhoff normal forms, numerical continuation, and singular reduction of resonant normal forms. In nonresonant regions, Arnold's theorem is applied when the quartic nondegeneracy condition holds, while sixth-order normalization resolves the degenerate cases. The $2$:$-1$ and $3$:$-1$ resonances are analyzed using the theorems of Alfriend, Markeev, and Meyer, including a classification of the $3$:$-1$ resonant dynamics. Singular reduction provides the corresponding reduced orbit spaces and reveals saddle-center bifurcations associated with the appearance and disappearance of periodic-orbit families. These results extend previous stability analyses beyond symmetric mass configurations and provide a geometric description of the resonant dynamics in the unequal-mass equilateral restricted four-body problem.
