Automatic Identification Of Scale-free Intervals For The Largest Lyapunov Exponent And Correlation Dimension Of Chaotic Time Series

Chaos and fractal are important nonlinear science branches. They have been ap-plied in astrophysics, medicine, chemistry, computers, etc, and have broad application prospects. With the rapid development of computers, using numerical methods to char-acterize the chaotic characteristics becomes possible. In particular, calculating the largest Lyapunov exponent and the correlation dimension accurately from time series is very im-portant. At present, the small data method for calculating the largest Lyapunov exponent and the G-P method for computing correlation dimension are the mainstream methods. However, in these methods, artificial selection of scale-free interval results in inaccurate calculations. In this dissertation, the problem of automatically selecting the scale-free interval is studied. A new method is proposed and applied to the study of solar activity. The main innovations and achievements are as follows:1. In the computation of the largest Lyapunov exponent by the small data method, a method based on fuzzy C-means clustering is proposed to reduce the error of iden-tifying the scale-free interval caused by human factors. It identifies the scale-free interval by machine learning according to the changing characteristic of average divergence index curve. Firstly, the average divergence index data are calculated from the small data method for chaotic time series. Secondly, the fuzzy C-means clustering method is used for dividing the data into two classes, and the unsaturated data are retained. Thirdly, the second order difference of retained data is classified, where the near zero data is obtained and the gross error is eliminated. Then the scale-free interval is identified by using statistical methods from the retained ef-fective data. Finally, the largest Lyapunov exponent is obtained by using the least square method to fit the points of the scale-free interval. By simulation, the results obtained by the new method are close to the reference values. In order to improve the calculation accuracy of the method, we use the intelligent algorithm to optimize the aboved method, which is more accurate than known methods in the most cas-es. But in terms of computational efficiency, the method based on fuzzy C-means clustering algorithm is the best.2. The idea of the above method is applied to calculate the correlation dimension of the chaotic time series, in order to reduce the error caused by human for identi-fy the scale-free interval. It identifies the scale-free interval by machine learning according to the variation characteristics of nearly zero fluctuation of the second order derivative of the corresponding curve of the scale-free interval. Secondly, the simulated annealing genetic fuzzy C-means clustering method is used to classify the data, the zero fluctuation data was selected to retain, and then the gross errors were excluded from the selected data. Finally, the statistical analysis is carried out to identify the scale-free interval by using the best linearity degree. The new method is applied to simulate the two famous chaotic systems Lorenz and Henon The calculated results are in good agreement with the reference values. The ex-periments show that the proposed new method is more efficient and more accurate than subjective recognition. K-means based method and 2-means based method in identifying the scale-free interval.3. The proposed new methods and other nonlinear analysis techniques are used to study the solar activity in the long term time. The results showed that:(1) solar activity keeps the long term memory which is related to the past evolution; (2) solar activity shows a low-dimensional chaos, and thus it can only be predicted for a short-and-medium term:(3) sunspot areas have more complex changing rule than the sunspot numbers, which are consistent with their own physical properties:(4) for predicting solar activity, sunspots areas is more effective than sunspot numbers as a index of studying solar activity.4. The new method and other methods are combined to study the polar faculae and the sunspot numbers of two solar magnetic activity indices in the time interval from February 1952 to June 1998. By simulation examples, the following meaningful results are obtained:(1) the chaotic and fractal properties of solar activity are statis-tically different in the northern and southern hemispheres; (2) the chaotic behavior of solar activity at high latitudes is stronger than that at low latitudes. Furthermore, the high-latitude solar activity in the northern hemisphere has the most complex activities.