混沌时间序列分析与随机共振研究的信息论方法/Information Theoretic Approach to Chaotic Time Series Anal

2018-08-25 07:48:48

The signal 信号 method noise



混沌时间序列分析和随机共振是非线性动力学的两个重要研究课题,它们在物理、力学、生物等科学和工程领域的检测、信号处理、信息获取和利用中有着广泛的应用。本论文从信息论的角度对混沌时间序列分析和随机共振进行了深入的研究。论文大体分为两部分,第一部分包括第四、五、六章,分别研究了混沌伪随机序列的复杂度、时空混沌信号的耦合特性和混沌时间序列相空间重构参数的选取。第二部分包括第七、八、九章,这一部分研究了双稳系统的信号调制噪声效应及其数值分析、随机共振的量化分析。本论文的主要内容如下:
(1)从信息论的角度分析了混沌伪随机序列的复杂度,提出了用符号动力学的方法来分析混沌伪随机序列的复杂度,用符号熵作为判断序列复杂度大小的准则。以Logistic映射和耦合映射格子系统产生的混沌伪随机序列为例,说明了该方法的应用,并将计算结果与近似熵ApEn法的计算结果作了比较。结果表明,符号动力学方法可以有效地判断混沌伪随机序列的复杂度,而且比近似熵法更为优越。
(2)从信息论的角度研究了耦合映射格子系统的时空混沌行为,将符号分析法引入到时空混沌信号的动力学耦合特性分析中,提出了根据两个混沌信号之间的条件熵是否有一个尖锐而明显的极小值来确定它们之间的耦合关系,并用这一方法分析了耦合映射格子系统中时空混沌信号之间的耦合关系。结果表明,符号分析法可以有效地确定耦合映射格子系统不同格子混沌时序之间的耦合关系,而相关分析法则不能够有效地确定时空混沌信号的动力学耦合关系。
(3)根据信息论基本原理,研究了混沌时序相空间重构参数延迟时间和嵌入维数的选取问题,提出了确定相空间重构参数的信息论方法,将符号分析的方法引入互信息函数的计算,从而确定延迟时间。在此基础上,提出了根据重构向量的条件熵确定嵌入维数,给出了其原理和算法,通过对几种典型动力系统的数值计算,结果表明,该方法能够确定出合适的相空间重构嵌入维数。
(4)用解析和定性的方法研究了双稳随机共振系统的信号调制噪声效应,提出了双稳动力系统瞬时定态的概念,为分析双稳系统动力学特性提供了一种定性方法。研究结果表明,通过选择适当的系统参数,受正弦信号和白噪声共同作用的双稳系统中,不但可以发生随机共振,而且可以发生局域的和全局的信号调制噪声的效应。
(5)基于双稳系统的随机共振模型,对信号调制噪声效应进行了数值分析。结果表明,正弦信号在系统输出中的效应仍表现为一个正弦信号的形式,而白噪声在系统输出中的效应则表现为一个维纳过程的形式。通过选择合适的系统参数,可以大为减小系统输出中信号和噪声之间的耦合效应,这样系统可以很大地抑制噪声,从而在双稳系统输出信号中产生信号调制噪声效应。利用这一结果,还研究了微弱正弦信号的检测问题。研究表明,选择较小的系统参数,使双稳系统产生信号调制噪声效应,再通过对系统输出信号作功率谱分析可以较精确地估计出微弱正弦信号的频率和幅值,而且还可用于多个微弱正弦信号的检测。
(6)从信息论的观点研究了双稳系统在正弦信号和白噪声作用下的随机共振。通过将系统输出信号和输入正弦信号采样离散化为时间序列,再用粗粒化方法将它们转化为相应的符号序列,计算出输出信号序列与输入正弦信号序列间的互信息,提出用输出信号序列熵归一化后的互信息来刻画系统输出信号对输入正弦信号的同步共振程度。数值仿真表明,表征双稳系统输入和输出间的同步共振程度的归一化互信息对噪声强度表现出一种峰状曲线,这一信息指标可以用来刻画双稳系统的随机共振。


Chaotic time series analysis and stochastic resonance are two important research subjects in nonlinear dynamics. There are many applications of chaotic time series analysis and stochastic resonance in detection, signal processing, information acquisition and utilization in various areas of science and engineering, such as in physics, mechanics, biology, and so on. In this dissertation these two subjects are studied in depth from information theoretical point of view. The dissertation may be broadly divided into two parts. The first part, comprising Chapters 4, 5 and 6, discusses the complexity analysis of chaotic pseudo-random sequences, the dynamical coupling of spatiotemporally chaotic signals, and the determination of delay time and embedding dimension for phase space reconstruction of chaotic time series, respectively. The second part of the dissertation comprising Chapters 7, 8 and 9, discusses the effect of signal modulating noise in bistable stochastic resonance systems, the numerical analysis of this effect, and the quantification of stochastic resonance, respectively. The main contributions of the dissertation are as follows:
The complexity of chaotic pseudo-random sequences is studied from information theoretical point of view. The symbolic dynamics method is presented for the complexity analysis of chaotic pseudo-random sequences, and symbolic entropy is presented as the criterion of the complexity. The method is applied to the cases of Logistc map and one-way coupled map lattices to demonstrate how it works, and a comparison is made between it and the approximate entropy method. The results show that this method is applicable to distinguish the complexities of different chaotic pseudo-random sequences, and it is superior to the approximate entropy method.
The dynamical coupling of spatiotemporally chaotic signals is studied from information theoretical point of view. The symbolic analysis method is presented to investigate the dynamical coupling of spatiotemporally chaotic signals, and the existence of a sharp minimum of the conditional entropy of two chaotic time series as a function of the shift parameter is considered as an indication whether or not there exists a mutual relation between the two time series. The method is used to investigate the dynamical coupling of spatiotemporally chaotic signals produced by differernt lattices of the coupled map lattice models. The results show that the symbolic analysis method is applicable for manifesting the dynamical coupling of spatiotemporally chaotic signals, whereas the correlation function is inadequate for this purpose.
The determination of delay time and embedding dimension for phase space reconstruction of chaotic time series is studied based on information theory. An information theoretic method is presented to determine the parameters of phase space reconstruction. The symbolic analysis method is introduced to compute mutual information for determining delay time. Further, it is presented that the embedding dimension be determined by considering the variation of the conditional entropy of the reconstructed vector with the dimension. The method is used in the numerical analysis of four typical dynamical systems. The numerical simulations verify that the method is applicable for determining an appropriate embedding dimension.
The effect of signal modulating noise in bistable stochastic resonance systems is studied analytically and qualitatively. The concept of instantaneous steady state is proposed for bistable dynamical systems, with which the response mechanism of bistable systems to the driving signal has been analyzed. The investigation shows that not only stochastic resonance but also the local and global effect of signal modulating noise can manifest in bistable systems driven by both a weak sinusoidal signal and white noise when the parameters are properly selected.
The effect of signal modulating noise is studied numerically based on the stochastic resonance model of bistable systems. The numerical analysis reveals that the effect of an input sinusoidal signal is also a sinusoidal signal in the output, and the effect of white noise is a Wiener process. When the system parameters are small enough the coupling effect between the sinusoidal signal and white noise can be largely weakened. In this condition the noise can be largely reduced in the system output, the effect of signal modulating noise can occur in bistable stochastic dynamical systems. The result is used in the detection of weak sinusoidal signal. The investigation shows the frequency and the magnitude of a weak sinusoidal signal submerged in noise can be evaluated by making an analysis on the power spectral density of the system output signal when the effect of signal modulating noise occurred by selecting the system parameters appropriately. The numerical simulation shows the application is also effective on the detection of several weak sinusoidal signals submerged in noise.
Stochastic resonance in bistable systems driven by a sine signal and white noise is studied from information theoretical point of view. By converting the input and out signals into discrete time series using the coarse-grained method, the mutual information between the output signal and the input sine signal is calculated with the symbolic analysis method, the mutual information which is normalized by the entropy of the output signal is presented to quantify the synchronous resonance between the output signal and the input sine signal. Numerical simulation results show that this measure undergoes resonance-like behavior as a function of the noise level, and can be used to quantify stochastic resonance in bistable systems.