非线性动力系统长度分形维理论及工程应用研究

在Hausdorff维和分配维的基础上，将超立方体作为覆盖单元，根据非线性动力系统振动波形的前向性特点和拓扑映射技术，提出了长度分形维的概念和计算公式．对长度分形维的数学严密性进行证明并对其工程应用可靠性作出评价，说明其具有概念明晰、计算简单、对理论值逼近较快、适合于少样本点集等特点．工程应用实例表明。基于长度分形维能够对非线性动力系统进行状态空间检测，噪声免疫力强和缺陷识别精度较高是其突出特色；它对理论分形维的适度近似性使其可以作为非线性动力系统的有效特征量，依其进行模式辨识是可行和有效的。

Theory and engineering application of length fractal dimension on nonlinear vibration waveform

Using supercube as overlying cell, the fractal dimension computing method to waveform proposed by Katz is analyzed and its shortage is pointed out. Based on Hausdorff and divider dimensions and using vibration waveform's property of going forward and topology map,the definition and formula of length fractal dimension (D _L) are formed. Its strictness in math is proved and reliability in engineering application is assessed, so the conclusion is made that D _L is of characteristics of concept clear, computation simple, quick to the theory solution and suitable to the small data. It is proved by practical application that state space inspection of nonlinear dynamic system can be carried out through D _L and its distinctions of strong immunity to noise and high accuracy to fault detection are demonstrated. D _L approximates properly to theoretic value makes it an effective characteristic quantity of nonlinear dynamic system by which mode identification is done well.