In this study, we discuss a relationship between the behavior of nonlinear dynamical systems and geometry of a system of second-order differential equations based on the Jacobi stability analysis. We consider how a maximal Lyapunov exponent is related to the geometric quantities. As a result of a theoretical investigation, the maximal Lyapunov exponent can be represented by a nonlinear connection and a deviation curvature. Thus, this means that the Jacobi stability given by the sign of the deviation curvature affects the change of the maximal Lyapunov exponent. Additionally, for an equation of nonlinear pendulum, we numerically confirm the theoretical results. We observe that a change of the maximal Lyapunov exponent is related to a change of an average deviation curvature. These results indicate that the deviation curvature and Jacobi stability are essential for considering the change of maximal Lyapunov exponent.

中文翻译：

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基于偏差曲率的最大李雅普诺夫指数几何解释

在这项研究中，我们讨论了非线性动力系统的行为与基于雅可比稳定性分析的二阶微分方程系统的几何之间的关系。我们考虑最大李雅普诺夫指数如何与几何量相关。作为理论研究的结果，最大李雅普诺夫指数可以用非线性连接和偏差曲率来表示。因此，这意味着由偏差曲率的符号给出的雅可比稳定性会影响最大李雅普诺夫指数的变化。此外，对于非线性摆方程，我们在数值上证实了理论结果。我们观察到最大李雅普诺夫指数的变化与平均偏差曲率的变化有关。