皮层脑电时间序列的相空间重构及非线性特征量的提取

对麻醉的SD大鼠在癫痫发作前后两种状态的皮层脑电(ECoG)的时间序列,用多种有效的方法和分析技术,使得大量的ECoG时间序列得以正确的分析,并得出重要的结论.首先利用延时坐标法重构吸引子;计算互信息函数,取互信息函数第一次达到最小值的延时为重构延时时间,提出将伪邻点法和Cao法相结合的方法确定最佳嵌入维数.然后采用非线性预报和替代数据法相结合的方法确定ECoG为混沌时间序列,从不同角度得出了ECoG不是低维混沌的结论.在ECoG相空间重构的基础上,计算了最大Lyapunov指数(LLE).应用了近似熵这一标量对ECoG进行刻画,计算结果表明:癫痫发作前的皮层脑电的最大Lyapunov指数和近似熵都明显地高于癫痫发作后的,这可能为理解癫痫发病机理,预报癫痫发作和治疗提供一定的思路. 关键词： 皮层脑电 互信息 伪邻点法 最大Lyapunov指数 近似熵     

确定延迟时间互信息法的一种算法

介绍了相空间重构中确定重构延迟时间的互信息法理论及其具体的计算方法.针对互信息法应用中计算复杂、程序难以编写问题,提出了一种基于网格层数的简便实用程序算法.对网格层数参数进行了结果分析,表明在利用互信息法确定重构延迟时间时,只需要计算到某一个网格层数即可,不需要计算出精确的互信息值,简化了计算的复杂程度.最后通过对Lorenz和Rossler两个吸引子的Lyapunov指数计算,证实了该算法的合理性.     

The analysis of a novel 3-D autonomous system and circuit implementation

This Letter presents a new three-dimensional autonomous system with four quadratic terms. The system with five equilibrium points has complex chaotic dynamics behaviors. It can generate many different single chaotic attractors and double coexisting chaotic attractors over a large range of parameters. We observe that these chaotic attractors were rarely reported in previous work. The complex dynamical behaviors of the system are further investigated by means of phase portraits, Lyapunov exponents spectrum, Lyapunov dimension, dissipativeness of system, bifurcation diagram and Poincaré map. The physical circuit experimental results of the chaotic attractors show agreement with numerical simulations. More importantly, the analysis of frequency spectrum shows that the novel system has a broad frequency bandwidth, which is very desirable for engineering applications such as secure communications.     

Chaotic Phenomena in a Two-Photon Laser with Injected Signal

The instability and the chaotic phenomena in a two-photon laser with injected signal are discussed for the homogeneously broadened single mode ring cavity. The structure of the system's attractors is considered by using the Lyapunov exponents and the Lyapunov dimension. The strange attractors of chaos and superchaos are found. The strange attractor displaying superchaos is not observed in one-photon laser with injected signal.     

Computing the pressure for Axiom-A attractors by time series and large deviations for the Lyapunov exponent

For the Axiom-A attractors a relation is given between the topological pressure and the spectrum of the generalized Lyapunov exponents. As a consequence, a simple formula is found to compute the topological entropy of the attractor by means of a time series. The results are used to compute the large deviations for positive Lyapunov exponents.     

Liu混沌系统的混沌分析及电路实验的研究

研究了一种新型混沌系统——Liu混沌系统的基本动力学行为以及电路实现的问题，给出了相图、庞卡莱映射、功率谱以及李雅普诺夫指数，基于李雅普诺夫指数谱和分叉图分析了系统参数对Liu混沌系统的影响.最后设计硬件电路证实了Liu混沌系统以及Liu混沌系统随系统参数变化时的各种状态的存在.给出数值仿真和电路实验的结果. 关键词： Liu混沌系统 分岔 电路实验     

Lyapunov exponent calculation of a two-degree-of-freedom vibro-impact system with symmetrical rigid stops

A two-degree-of-freedom vibro-impact system having symmetrical rigid stops and subjected to periodic excitation is investigated in this paper. By introducing local maps between different stages of motion in the whole impact process, the Poincar'e map of the system is constructed. Using the Poincar'e map and the Gram-Schmidt orthonormalization, a method of calculating the spectrum of Lyapunov exponents of the above vibro-impact system is presented. Then the phase portraits of periodic and chaotic attractors for the system and the corresponding convergence diagrams of the spectrum of Lyapunov exponents are given out through the numerical simulations. To further identify the validity of the aforementioned computation method, the bifurcation diagram of the system with respect to the bifurcation parameter and the corresponding largest Lyapunov exponents are shown.     

Continuity of the Lyapunov Exponents for Quasiperiodic Cocycles

Consider the Banach manifold of real analytic linear cocycles with values in the general linear group of any dimension and base dynamics given by a Diophantine translation on the circle. We prove a precise higher dimensional Avalanche Principle and use it in an inductive scheme to show that the Lyapunov spectrum blocks associated to a gap pattern in the Lyapunov spectrum of such a cocycle are locally Hölder continuous. Moreover, we show that all Lyapunov exponents are continuous everywhere in this Banach manifold, irrespective of any gap pattern in their spectra. These results also hold for Diophantine translations on higher dimensional tori, albeit with a loss in the modulus of continuity of the Lyapunov spectrum blocks.     

Spatiotemporal chaos and noise

Low-dimensional chaotic dynamical systems can exhibit many characteristic properties of stochastic systems, such as broad Fourier spectra. They are distinguishable from stochastic processes through finite values for their dimension, Lyapunov exponents, and Kolmogorov-Sinai entropy. We discuss how these characteristic observables are modified in spatiotemporal chaotic systems like. coupled map lattices. We analyze with the help of Lyapunov concepts how the stochastic limit is approached and how these properties can be observed directly through local dimension measurements from reconstructed time series. Finally, we discuss the interaction of spatiotemporal attractors with external noise and possible connections to problems of pattern selection and stability.     

Dynamical behaviors of a chaotic system with no equilibria

Based on Sprott D system, a simple three-dimensional autonomous system with no equilibria is reported. The remarkable particularity of the system is that there exists a constant controller, which can adjust the type of chaotic attractors. It is demonstrated to be chaotic in the sense of having a positive largest Lyapunov exponent and fractional dimension. To further understand the complex dynamics of the system, some basic properties such as Lyapunov exponents, bifurcation diagram, Poincaré mapping and period-doubling route to chaos are analyzed with careful numerical simulations.     

强迫布鲁塞尔振子奇异吸引子的柯尔莫哥洛夫容量和李雅普诺夫维数

本文用周期采样降低吸引子维数的办法,计算和讨论了强迫布鲁塞尔振子的几种典型吸引子的柯尔莫哥洛夫容量d_C和李雅普诺夫维数d_L。结果表明,人们关于d_C和d_L关系的推测是正确的。这些推测是:当最大李雅普诺夫指数λ₁>0时d_C=d_L关系成立,当λ₁=0时d_C=d_L关系不一定成立。文中指出和论证了强迫布鲁塞尔振子的Runge-Kutta差分方程使上述算法给出的d_C不收敛的原因和克服办法。文中还指出:可能是类似的微分方程的差分化的原因,使得用单个观察量的时间系列来计算容量的办法遇到了发散困难。 关键词：     

Transition from phase to generalized synchronization in time-delay systems

The notion of phase synchronization in time-delay systems, exhibiting highly non-phase-coherent attractors, has not been realized yet even though it has been well studied in chaotic dynamical systems without delay. We report the identification of phase synchronization in coupled nonidentical piecewise linear and in coupled Mackey-Glass time-delay systems with highly non-phase-coherent regimes. We show that there is a transition from nonsynchronized behavior to phase and then to generalized synchronization as a function of coupling strength. We have introduced a transformation to capture the phase of the non-phase-coherent attractors, which works equally well for both the time-delay systems. The instantaneous phases of the above coupled systems calculated from the transformed attractors satisfy both the phase and mean frequency locking conditions. These transitions are also characterized in terms of recurrence-based indices, namely generalized autocorrelation function P(t), correlation of probability of recurrence, joint probability of recurrence, and similarity of probability of recurrence. We have quantified the different synchronization regimes in terms of these indices. The existence of phase synchronization is also characterized by typical transitions in the Lyapunov exponents of the coupled time-delay systems.     

Dynamics of a three DOF mechanical system with dry friction

The dynamics of a three-block mechanical system is investigated: each block is pulled by a belt and is subjected to linear elastic and nonlinear frictional forces which induce oscillations in the system. The study of the full dynamics of the system is partially reduced to the study of a two-dimensional map; its attractors, their basins of attraction and their Lyapunov exponents provide a powerful tool to understand the dynamic behaviour of the full mechanical system which possesses rich dynamics characterised by periodic, quasi-periodic, chaotic and hyper-chaotic attractors.     

Hyperchaotic qualities of the ball motion in a ball milling device

Ball collisions in milling devices are governed by complex dynamics ruled by impredictable impulsive forces. In this paper, nonlinear dynamics techniques are employed to analyze the time series describing the trajectory of a milling ball in an empty container obtained from a numerical model. The attractor underlying the system dynamics was reconstructed by the time delay method. In order to characterize the system dynamics the calculation of the spectrum of Lyapunov exponents was performed. Six Lyapunov exponents, divided into two terns with opposite sign, were obtained. The detection of the positive tern demonstrates the occurrence of the hyperchaotic qualities of the ball motion. A fractal Lyapunov dimension, equal to 5.62, was also obtained confirming the strange features of the attractor. (c) 1999 American Institute of Physics.     

Dynamics and Entropy in the Zhang Model of Self-Organized Criticality

We give a detailed study of dynamical properties of the Zhang model, including evaluation of topological entropy and estimates for the Lyapunov exponents and the dimension of the attractor. In the thermodynamic limit the entropy goes to zero and the Lyapunov spectrum collapses.     

Period-Doubling Cascades and Strange Attractors in Extended Duffing-Van der Pol Oscillator

The dynamical behavior of the extended Duffing-Van der Pol oscillator is investigated numerically in detail. With the aid of some numerical simulation tools such as bifurcation diagrams and Poinearé maps, the different routes to chaos and various shapes of strange attractors are observed. To characterize chaotic behavior of this oscillator system, the spectrum of Lyapunov exponent and Lyapunov dimension are also employed.     

Nonlinear analysis using Lyapunov exponents in breast thermograms to identify abnormal lesions

Breast diseases are one of the major issues in women’s health today. Early detection of breast cancer plays a significant role in reducing the mortality rate. Breast thermography is a potential early detection method which is non-invasive, non-radiating, passive, fast, painless, low cost, risk free with no contact with the body. By identifying and removing malignant tumors in early stages before they metastasize and spread to neighboring regions, cancer threats can be minimized. Cancer is often characterized as a chaotic, poorly regulated growth. Cancerous cells, tumors, and vasculature defy have irregular shapes which have potential to be described by a nonlinear dynamical system. Chaotic time series can provide the tools necessary to generate the procedures to evaluate the nonlinear system. Computing Lyapunov exponents is thus a powerful means of quantifying the degree of the chaos.In this paper, we present a novel approach using nonlinear chaotic dynamical system theory for estimating Lyapunov exponents in establishing possible difference between malignant and benign patterns. In order to develop the algorithm, the first hottest regions of breast thermal images are identified first, and then one dimensional scalar time series is obtained in terms of the distance between each subsequent boundary contour points and the center of the mass of the first hottest region. In the next step, the embedding dimension is estimated, and by time delay embedding method, the phase space is reconstructed. In the last step, the Lyapunov exponents are computed to analyze normality or abnormality of the lesions. Positive Lyapunov exponents indicates abnormality while negative Lyapunov exponents represent normality. The normalized errors show the algorithm is satisfactorily, and provide a measure of chaos. It is shown that nonlinear analysis of breast thermograms using Lyapunov exponents may potentially capable of improving reliability of thermography in breast tumor detection as well as the possibility of differentiating between different classes of breast lesions.     

Modelling of chaotic systems based on modified weighted recurrent least squares support vector machines

Positive Lyapunov exponents cause the errors in modelling of the chaotic time series to grow exponentially. In this paper, we propose the modified version of the support vector machines (SVM) to deal with this problem. Based on recurrent least squares support vector machines (RLS-SVM), we introduce a weighted term to the cost function tocompensate the prediction errors resulting from the positive global Lyapunov exponents. To demonstrate the effectiveness of our algorithm, we use the power spectrum and dynamic invariants involving the Lyapunov exponents and the correlation dimension as criterions, and then apply our method to the Santa Fe competition time series. The simulation results shows that the proposed method can capture the dynamics of the chaotic time series effectively.     

The analysis of complex behaviours of a novel three dimensional autonomous system

This paper introduces a new three dimensional autonomous system with five equilibrium points.It demonstrates complex chaotic behaviours within a wide range of parameters,which are described by phase portraits,Lyapunov exponents,frequency spectrum,etc.Analysis of the bifurcation and Poincar’e map is used to reveal mechanisms of generating these complicated phenomena.The corresponding electronic circuits are designed,exhibiting experimental chaotic attractors in accord with numerical simulations.Since frequency spectrum analysis shows a broad frequency bandwidth,this system has perspective of potential applications in such engineering fields as secure communication.     

Global aspects of turbulence induced by heteroclinic cycles in competitive diffusion Lotka-Volterra equation

This paper presents detailed numerical results of the competitive diffusion Lotka-Volterra equation (May-Leonard type). First, we derive the global phase diagrams of attractors in the parameter space including the system size, where transition lines between simple attractors are clearly obtained in accordance with the results of linear stability analysis, but the transition borders become complex when multi-basin structures appear. The complex aspects of the transition borders are studied in the case when the system size decreases. Next, we show the statistical aspects of the turbulence with special attention to the onset of the supercritical Hopf bifurcation. Several characteristic quantities, such as correlation length, correlation time, Lyapunov spectra and Lyapunov dimension, are investigated in detail near the onset of turbulence. Our data show the critical scaling law near the onset only in the restricted parameter domain. However even when the critical indices are not determined accurately, it is shown that the empirical scaling relations are obtained in a wide parameter domain far from the onset point and those scaling indices satisfy several relations. These scaling relations are discussed in comparison with the result derived by the phase reduction method. Lastly, we make a conjecture about the stability of an ecosystem based on the bifurcation diagram: the ecosystem obeying the Lotka-Volterra equation in the case of May-Leonard type is stabilized more as the system size increases.