In this paper, the Jacobi stability of a two-degree-of-freedom mechanical system is studied by the innovative application of KCC-theory, namely differential geometric methods. We discuss the Jacobi stability of two equilibria and a periodic orbit by constructing geometric invariants. Both the regions of Jacobi stability and Lyapunov stability are presented to show the difference. We draw the phase portraits of the deviation vector near two equilibria under specific parameter values and initial conditions, and point out the sensitivity of deviation vector to initial conditions. In addition, the corresponding instability exponent and curvature are applicable for predicting the onset of chaos, which help us to detect chaotic behaviors quantitatively.

中文翻译：

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雅可比稳定性分析和两自由度机械系统中的混沌发生

本文通过创新应用KCC理论，即微分几何方法，研究了二自由度机械系统的雅可比稳定性。我们通过构造几何不变量来讨论两个平衡和一个周期轨道的雅可比稳定性。Jacobi 稳定区域和 Lyapunov 稳定区域均被呈现以显示差异。我们在特定的参数值和初始条件下绘制了两个平衡点附近的偏差矢量的相位图，指出了偏差矢量对初始条件的敏感性。此外，相应的不稳定性指数和曲率可用于预测混沌的发生，这有助于我们定量地检测混沌行为。