高维非线性动力系统降维理论综述

Structures and mechanisms in the engineering field possess characteristics such as high dimensions, nonlinearity, and strong coupling, leading to complex dynamic behaviors. In the related research field, the dimensionality reduction method is of great significance for the study of high-dimensional complex nonlinear dynamical systems. These methods can reduce the complexity of data, overcome the “curse of dimensionality” in dynamical systems, and improve computational efficiency. They can also compress and reconstruct the characteristics of high-dimensional data, extracting core characteristics to better reveal its inherent laws and features. Furthermore, they can simplify models, reduce model complexity, and improve model stability and interpretability. In recent years, the dimension reduction method system has gradually developed and improved, and many scholars have utilized them to achieve theoretical research on high-dimensional complex systems. Based on this, this paper summarizes the dimension reduction theory for nonlinear high-dimensional systems. It focuses on introducing the basic ideas, application status, and advantages and disadvantages of dimension reduction methods such as Central Manifold Theorem dimension reduction method, Lyapunov-Schmidt method, Proper Orthogonal Decomposition method (POD), and nonlinear Galerkin method. Additionally, it briefly introduces the application of other dimension reduction methods in practical problems. Finally, aiming at the problems existing in current dimension reduction methods, it proposes possible improvement plans and prospects for future research directions.