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

Abstract

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, Poincar 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.