混沌时间序列中最大Lyapunov指数与关联维数的无标度区间自动识别研究

发布时间：2018-07-31 20:37

【摘要】：混沌与分形是非线性科学中一个重要的分支,其已经渗透到天体物理、医学、化学、计算机等各个学科,具有广泛的应用前景。随着计算机的快速发展,数值方法刻画混沌特征成为可能,其中从时间序列准确计算最大Lyapunov指数和关联维数尤为重要。目前,计算最大Lyapuno v指数的小数据量方法和计算关联维数的G-P方法是其中最主流的方法,但是在计算过程中人为选择无标度区间,导致计算结果不准确,因此本文对无标度区间自动识别问题进行了研究,提出了新方法,并将该方法应用到太阳长期活动规律的研究,取得了一些有意义的结果。本文的创新点如下：一、在小数据量方法计算最大Lyapunov指数的过程中,为了减少人为因素识别无标度区间带来的误差,提出一种基于模糊C均值聚类的方法,该方法根据平均发散程度指数曲线的变化特征,利用模糊C均值聚类进行识别。首先,利用小数据量方法对混沌时间序列进行计算得到平均发散程度指数集合。其次,利用模糊C均值聚类算法对平均发散程度指数集合进行分类,保留不饱和数据。然后,对不饱和的二阶差分数据进行分类,得到零附近波动数据并剔除粗大误差,再对保留的有效数据利用统计方法识别出无标度区间。最后,对无标度区间对应的点进行最小二乘法拟合得到最大Lyapunov指数。通过实例仿真,新方法所得结果接近参考值。为了提高该方法的计算精度,利用模拟退火算法与遗传算法对其优化,优化后的方法比已知方法,在大多数情况下,计算精度更高,但是在计算效率方面,基于模糊C均值聚类的新方法表现最好。二、将上述方法的思想应用于计算混沌时间序列的关联维数,用来减少人为因素识别无标度区间带来的误差。该方法根据无标度区间对应曲线的二阶导数在零附近波动的变化特征,利用分类算法进行识别。首先计算双对数关联积分的二阶差分,然后利用模拟退火遗传模糊C均值聚类方法对该数据进行分类,选出在零附近波动的数据,再剔除粗大误差保留有效数据,最后进行统计分析识别出线性度最好的作为无标度区间。应用新方法对两个著名的混沌系统Lorenz 和 Henon进行了仿真,计算结果与参考值非常吻合。实验表明,所提出的新方法具有一定的抗噪性,与主观识别、基于K-means方法和基于2-means方法比较,’可以更加精确的识别无标度区间。三、将新方法以及其它非线性分析技术应用到太阳的长期活动规律研究中,研究结果表明：(1)太阳长期活动趋势与过去的演化过程是密切相关,具有长期记忆性；(2)太阳长期活动展示着低维混沌,由于其混沌的特征,只能进行中短期的太阳活动预报；(3)太阳黑子面积的变化规律比太阳黑子数更复杂,这与它们本身的物理意义相符合；(4)太阳黑子面积相对于太阳黑子数作为研究太阳活动的指标更有效。四、将新方法与其它方法结合对1952年2月至1998年6月的极区光斑和太阳黑子数的两种太阳活动指标进行研究,通过实例仿真,得出了一些有意义的结果：(1)南北半球的太阳活动的混沌与分形特征存在统计上的差异性；(2)太阳高纬度活动现象比低纬度活动现象具有更强的混沌程度和更复杂的分形结构,其中北半球的高纬度太阳活动规律最复杂。
[Abstract]:Chaos and fractal are an important branch of nonlinear science. It has penetrated into various disciplines such as astrophysics, medicine, chemistry, computer and so on. It has extensive application prospects. With the rapid development of the computer, it is possible to describe the chaotic characteristics by numerical methods, in which the maximum Lyapunov index and the correlation dimension are accurately calculated from the time series. It is particularly important. At present, the calculation method of the maximum Lyapuno V index and the G-P method for calculating the correlation dimension are the most mainstream methods. However, in the calculation process, the artificial selection of the scale-free interval, which leads to the inaccuracy of the calculation results, is carried out in this paper. A new method is proposed and a new method is put forward. This method is applied to the study of the law of the long-term activity of the sun, and some meaningful results have been obtained. The innovation points of this paper are as follows: first, in the process of calculating the maximum Lyapunov exponent of small amount of data, a method based on fuzzy C mean clustering is proposed in order to reduce the error caused by the identification of the scale free interval of human factors. According to the variation characteristics of the average divergence index curve, the fuzzy C means clustering is used to identify. First, the mean divergence index set is obtained by using the small data quantity method to calculate the chaotic time series. Secondly, the fuzzy C mean clustering algorithm is used to classify the average divergence index set, and the unsaturated data is retained. After that, we classify the two order differential data of the unsaturated zone, get the wave data near zero and eliminate the rough error, and then identify the scale-free interval for the reserved effective data using statistical methods. Finally, the least square fitting is used to get the maximum Lyapunov exponent of the points corresponding to the scale-free interval. It is close to reference value. In order to improve the calculation accuracy of this method, the simulated annealing algorithm and genetic algorithm are used to optimize it. The optimization method is more accurate than the known method in most cases, but in the calculation efficiency, the new method based on fuzzy C means clustering is the best. Two, the idea of the above method is applied to the method. The correlation dimension of the chaotic time series is calculated to reduce the error caused by the artificial factor identification of the scale-free interval. The method is identified by the classification algorithm based on the variation characteristics of the fluctuation of the two order derivative of the scale-free interval corresponding to zero. First, the two order difference of the double logarithmic correlation integral is calculated, and then the simulated annealing remains are used. The fuzzy C means clustering method is used to classify the data, select the data that fluctuate near zero, and then remove the gross error and retain the effective data. Finally, the statistical analysis is used to identify the best line free degree as a scale-free interval. The new method is used to simulate the two famous chaotic systems, Lorenz and Henon, and the results and reference are calculated. The experimental results show that the proposed method has a certain noise resistance, and the subjective recognition, based on the K-means method and the 2-means based method, can more accurately identify the scale-free interval. Three, the new method and other nonlinear analysis techniques are applied to the study of the long-term activity law of the sun, and the results show that the results show that the new method and other nonlinear analysis techniques are applied to the study of the long-term activity of the sun. (1) the long-term activity trend of the sun is closely related to the evolution process of the past, and has long memory. (2) the long-term activity of the sun shows low dimensional chaos. Because of its chaotic characteristics, the solar activity can only be predicted in the middle and short term; (3) the variation of sunspot area is more complex than the number of sunspots, which is related to the physics of their own. The significance is conformed; (4) the area of sunspot area is more effective than the sunspot number. Four, the new method and other methods are combined to study two kinds of solar activity indexes of the polar spot and sunspot number from February 1952 to June 1998, and some meaningful results are obtained through the simulation of the case. (1) there is a statistical difference between the chaotic and fractal characteristics of the solar activity in the northern and southern hemispheres; (2) the high latitude activity of the sun is more chaotic and more complex than the low latitude activity, and the high latitude solar activity in the northern hemisphere is the most complex.
【学位授予单位】：中国科学院重庆绿色智能技术研究院
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O211.61

【参考文献】