analytic automorphism operator

例句：

1. 1. In the study of complexsystems, the analytic automorphism operatoris a crucial tool for analyzing the dynamicsof these systems.
   在研究复杂系统时，分析自同构算子是分析这些系统动态的关键工具。
2. 2. The analyticautomorphism operator has been used to studythe behavior of various types of nonlinear systems.
   分析自同构算子已被用于研究各种类型的非线性系统的动态行为。
3. 3. Theanalytic automorphism operator has proven to bean effective tool in the analysis of dynamical systems with multiple degrees of freedom.
   分析自同构算子已证明是一种有效的工具，用于分析具有多个自由度的动力学系统。