动力系统的几何与稳定性
Geometry and stability of dynamical systems
摘要：
我们从几何的角度重新考虑了连续演化动力系统解的全局稳定性和局部稳定性。我们澄清，一个明确的定义的稳定性一般需要选择额外的几何结构，这不是固有的动力系统本身。而全球李雅普诺夫稳定性的基础上选择的扰动向量束的范数，我们提出了一个定义的基础上选择的线性连接的局部稳定性。我们展示了这个定义如何再现已知的二阶动力系统的稳定性准则。与一般情况相反，拉格朗日系统的特殊几何提供了完全内在的全局和局部稳定性概念。我们证明，这些不遭受发生在与自然拉格朗日系统的莫佩屠斯-雅可比测地线分析的局限性。
英文摘要：
We reconsider both the global and local stability of solutions of continuously evolving dynamical systems from a geometric perspective. We clarify that an unambiguous definition of stability generally requires the choice of additional geometric structure that is not intrinsic to the dynamical system itself. While global Lyapunov stability is based on the choice of seminorms on the vector bundle of perturbations, we propose a definition of local stability based on the choice of a linear connection. We show how this definition reproduces known stability criteria for second order dynamical systems. In contrast to the general case, the special geometry of Lagrangian systems provides completely intrinsic notions of global and local stability. We demonstrate that these do not suffer from the limitations occurring in the analysis of the Maupertuis-Jacobi geodesics associated to natural Lagrangian systems.
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