首页|高维非线性动力系统降维理论综述

高维非线性动力系统降维理论综述

扫码查看
原文链接
  • 万方数据
  • 维普
工程领域中的结构和机构具有高维、非线性及强耦合等特性，导致其动态行为十分复杂。在相关研究领域中，降维方法对高维复杂非线性的动力学系统研究具有重要意义。它可以降低数据的复杂性，克服动力学系统的维数灾难，提高计算效率;也可以将高维数据的特征进行压缩和重构，提取出其核心特征，更好地揭示数据的内在规律和本质特征;还可以帮助简化模型，降低模型的复杂性，提高模型的稳定性和可解释性。近年来，降维方法体系逐渐发展完善，很多学者利用降维方法实现了高维复杂系统理论研究。基于此，针对非线性高维系统的降维理论进行了综述。重点介绍了基于中心流形理论的降维方法，Lyapunov-Schmidt方法，本征正交分解方法(Proper Orthogonal Decomposition)和非线性 Galerkin 方法等降维方法的基本思想、应用现状及各自的优缺点。此外，还简要介绍了实际问题中其他降维方法的应用。最后，针对现有降维方法存在的问题，提出了可能的改进方案和未来研究方向的展望。
Dimension Reduction Theory Review of High-dimensional Nonlinear Dynamical Systems
Structures and mechanisms in the engineering field possess characteristics such as high dimen-sions,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 com-plex 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 com-plexity,and improve model stability and interpretability.In recent years,the dimension reduction meth-od system has gradually developed and improved,and many scholars have utilized them to achieve theo-retical research on high-dimensional complex systems.Based on this,this paper summarizes the dimen-sion 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 Decom-position method(POD),and nonlinear Galerkin method.Additionally,it briefly introduces the applica-tion of other dimension reduction methods in practical problems.Finally,aiming at the problems exist-ing in current dimension reduction methods,it proposes possible improvement plans and prospects for future research directions.

dynamical systemsdimension reduction methodcentral manifoldLyapunov-Schmidt methodProper Orthogonal DecompositionGalerkin method

桑瑞涓、龚坚、路宽、靳玉林、张康宇、王衡

展开 >

西北工业大学 力学与土木建筑学院,西安 710000

中国人民解放军 91911 部队,三亚 572000

西南交通大学 机械工程学院,成都 611756

动力学系统 降维方法 中心流形 L-S方法 POD方法 Galerkin方法

国家自然科学基金资助项目国家自然科学基金资助项目KGJ 国防技术基础国家重点资助项目中央高校基本科研业务费

U224124312072263JSZL2022213 A001HYGJZN20232

2024

10.6052/1672-6553-2024-043

动力学与控制学报
中国力学学会　湖南大学

动力学与控制学报

CSTPCD
影响因子：0.446
ISSN：1672-6553
年,卷(期)：2024.22(9)