The Koopman Operator (KO) offers a promising alternative methodology to solve ordinary differential equations analytically. The solution of the dynamical system is analyzed in terms of observables, which are expressed as a linear combination of the eigenfunctions of the system. Coefficients are evaluated via the Galerkin method, using Legendre polynomials as a set of orthogonal basis functions. This tutorial provides a detailed analysis of the Koopman theory, followed by a rigorous explanation of the KO implementation in a computer environment, where a line-by-line description of a MATLAB code solves the Duffing oscillator application.

翻译：Koopman运算符(KO)为分析解决普通差分方程式提供了很有希望的替代方法。动态系统的解决方案以可观测性来分析,这些可观测性以系统元件的线性组合表示。通过Galerkin方法对系数进行评估,使用传说多语种作为一套正统基础功能。该指导对Koopman理论进行详细分析,然后对计算机环境中的KO实施情况作出严格解释,在计算机环境中,对MATLAB代码逐行描述可以解决数字振动器应用。