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
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.