Dynamic Mode Decomposition (DMD) is a popular data-driven analysis technique used to decompose complex, nonlinear systems into a set of modes, revealing underlying patterns and dynamics through spectral analysis. This review presents a comprehensive and pedagogical examination of DMD, emphasizing the role of Koopman operators in transforming complex nonlinear dynamics into a linear framework. A distinctive feature of this review is its focus on the relationship between DMD and the spectral properties of Koopman operators, with particular emphasis on the theory and practice of DMD algorithms for spectral computations. We explore the diverse "multiverse" of DMD methods, categorized into three main areas: linear regression-based methods, Galerkin approximations, and structure-preserving techniques. Each category is studied for its unique contributions and challenges, providing a detailed overview of significant algorithms and their applications as outlined in Table 1. We include a MATLAB package with examples and applications to enhance the practical understanding of these methods. This review serves as both a practical guide and a theoretical reference for various DMD methods, accessible to both experts and newcomers, and enabling readers to delve into their areas of interest in the expansive field of DMD.

翻译：动态模态分解（DMD）是一种流行的数据驱动分析技术，用于将复杂非线性系统分解为一组模态，通过谱分析揭示潜在的模式和动力学特性。本综述对DMD进行了全面且教学性的探讨，重点阐述了库普曼算子在将复杂非线性动力学转化为线性框架中的作用。本综述的一个显著特点是聚焦于DMD与库普曼算子谱特性之间的关系，尤其注重用于谱计算的DMD算法的理论与实践。我们探索了DMD方法的多样化"多重宇宙"，将其分为三大类：基于线性回归的方法、伽辽金逼近和结构保持技术。每类方法都针对其独特贡献和挑战进行了研究，并在表1中概述了重要算法及其应用的详细内容。我们提供了一套包含示例和应用的MATLAB工具包，以增强对这些方法的实践理解。本综述既作为各种DMD方法的实用指南，也作为理论参考，适用于专家和新手，使读者能够在广阔的DMD领域中深入探索其感兴趣的方向。