一种基于相邻数据依赖性的混沌分析方法

A chaotic analyzing method based on the dependence of neighbor sub-sequences in the data series

混沌作为一种复杂的非线性行为,广泛存在于各个行业领域,对于混沌的研究具有重要的理论意义和应用价值.现在常用的混沌分析方法,如Lyapunov指数、关联维数、Poincaré图等,需要解决相空间重构、线性标度区选取等问题,且不能很好地兼顾定性与定量分析两方面.基于此,提出一种度量相邻数据依赖性的混沌分析方法,通过计算相邻数据间的距离变化,将复杂的一维原始数据列转换为新的相邻距离值序列进行分析,避免了相空间重构等问题,对于不同的典型混沌模型,如Logistic模型、Duffing振子、Lorenz模型等,均具有较好的分析效果,能够描述不同模型的混沌特性,直观与量化分析效果均较好,且具有一定的抗噪能力,由于不需要掌握真实的模型信息,更适用于模型未知的复杂实际问题.将相邻数据的距离值对于不同混沌状态的区分作用应用于机械转子振动信号分析,可以明显地识别出转子工作状态的变化,表明该方法具有良好的实际应用前景和潜力.

Ever since the special characteristics hidden in the chaos was discovered, the chaotic behavior has been extensively studied as a ubiquitous and complex nonlinear dynamic phenomenon, which is gradually extending to various disciplines of natural and social science, and the significant values in the theoretical and the practical application have attracted much attention from scholars of different fields in the recent decades. Conventional methods of analyzing chaotic dynamic systems, including the Lyapunov exponent, correlation dimension, Poincaré map, unavoidably encounter some common problems, such as reconstruction of the phase space, determination of the linear area, etc. Besides, the current approaches each also possess a poor capability of balancing the direct observation and the quantitative calculation. Based on the fact that the neighbor data relate to each other to some degree, taking those shortages into consideration, aiming at depicting the chaotic features efficiently, a new method of analyzing the complicated chaotic motion is proposed. During the processing of that novel approach, the Euclidean distance is continuously computed to represent the dependence of the adjacent unit, after that, the original complicated array is converted into a simpler series composed of the distance of neighbor sub-sequences with more distinct characteristics. The mean value and the standard deviation of the newborn series are exacted to assist in describing the chaotic changing law. The method is adopted for studying the typical chaotic models, like Logistic model, Chebychev model, Duffing oscillator, Lorenz system, etc., which proves the good performances in explaining the chaotic variation rules in different systems. Based on the model verification, it could be seen that the method could detect the chaotic motion both qualitatively and quantitatively, and the ability for that method to resist the noise is improved up to some degree, what is more, the information about the real model is not required, thereby simplifying the analysis of the complicated chaotic behavior whose authentic model is unavailable. In addition, the method is applied to decomposing the vibration signal to monitor the working condition of the rotating rotor, and the results show that the conditional variation could be detected obviously. The analyses above show that the proposed method, on the basis of the dependence between nearby data, could perform well in observing the chaotic feature in an efficient way which simplifies the operation and clarifies the chaotic variation, moreover, the application potential of this method is worthy of great attention.