Analysis of convergence for initial condition estimation of coupled map lattices based on symbolic dynamics

A novel approach to the inverse problem of diffusively coupled map lattices is systematically investigated by utilizing the symbolic vector dynamics.The relationship between the performance of initial condition estimation and the structural feature of dynamical system is proved theoretically.It is found that any point in a spatiotemporal coupled system is not necessary to converge to its initial value with respect to sufficient backward iteration,which is directly relevant to the coupling strength and local mapping function.When the convergence is met,the error bound in estimating the initial condition is proposed in a noiseless environment,which is determined by the dimension of attractors and metric entropy of the system.Simulation results further confirm the theoretic analysis,and prove that the presented method provides the important theory and experimental results for better analysing and characterizing the spatiotemporal complex behaviours in an actual system.     

A method of estimating initial conditions of coupled maplattices based on time-varying symbolic dynamics

A novel computationally efficient algorithm in terms of the time-varying symbolic dynamic method is proposed to estimate the unknown initial conditions of coupled map lattices (CMLs). The presented method combines symbolic dynamics with time-varying control parameters to develop a time-varying scheme for estimating the initial condition of multi-dimensional spatiotemporal chaotic signals. The performances of the presented time-varying estimator in both noiseless and noisy environments are analysed and compared with the common time-invariant estimator. Simulations are carried out and the obtained results show that the proposed method provides an efficient estimation of the initial condition of each lattice in the coupled system. The algorithm cannot yield an asymptotically unbiased estimation due to the effect of the coupling term, but the estimation with the time-varying algorithm is closer to the Cramer--Rao lower bound (CRLB) than that with the time-invariant estimation method, especially at high signal-to-noise ratios (SNRs).     

基于符号动力学的耦合映像格子系统的初值估计

在符号动力学的基础上,深入探讨了基于动力学符号序列的局部耦合映像格子系统求逆问题.在理论上系统地分析耦合映像系统初值估计的性能与耦合系数及映射函数之间的数学关系,证明相空间I^M上的任意取值通过基于符号向量序列的逆迭代过程并不一定收敛至初值,其敛散性与耦合强度和映射函数的选择有直接关系.同时证明了混沌或其拓扑共轭的逆不一定为压缩映射,其总体的敛散性与整个逆迭代过程中的收敛与发散的强度对比有关.理论分析与数值实验结果完全一致,说明本文提出的耦合映像格子系统初值估计问题的分析 关键词： 耦合映像格子 符号动力学 初值估计     

基于符号向量动力学的耦合映像格子初始向量估计

在符号动力学的基础上,探讨了基于符号向量序列的局部耦合映像格子求逆问题,证明了相空间I^N上任意取值通过基于符号向量序列的逆迭代过程必然收敛至初始向量,提出了基于符号向量动力学的初始向量估计算法,从而建立了耦合映像格子符号序列和实际动力系统相空间的对应关系.实验结果表明,根据该算法可以有效建立符号向量序列和耦合映像格子相空间之间的对应关系,更好地刻画了实际模型的物理过程. 关键词： 耦合映像格子 符号动力学 初始向量估计     

基于符号向量动力学的耦合映像格子参数估计

本文,将符号动力学推广到耦合映像格子中,以Logistic映射下耦合映像格子为研究对象,研究控制参数对符号向量序列动力学特性的影响.通过研究耦合映像格子逆函数,给出耦合映像格子的遍历条件.进一步,将给出系统初始向量,禁止字以及控制参数的符号向量序列描述方法,并最终给出基于符号向量动力学的耦合映像格子控制参数估计方法.实验结果表明,根据本文算法可以有效建立符号序列和耦合映像格子控制参数之间的对应关系,能够更好地刻画了实际模型的物理过程. 关键词： 符号向量动力学 耦合映像格子 参数估计 遍历性     

一个全局耦合不连续映像格子中的冻结化随机图案模式

本文研究了一类既不连续又不可逆分段线性映像构成的全局耦合映像格子系统中的一类典型集体动力学行为, 即冻结化随机图案模式. 计算了平均同步序参量和最大李雅普诺夫指数随耦合强度的变化. 结果显示, 当耦合强度超过某个阈值后, 在给定动力学变量的初始下, 系统几乎都能达到完全或部分同步状态, 出现冻结化随机图案. 这些现象表明, 耦合映像格子系统中存在着多个共存的吸引子. 因此, 其冻结化图案的结构和分布敏感地依赖于格点动力学变量初始值的选取. 感兴趣地是, 即使当单映像处于混沌状态时, 格点间的耦合仍能将系统调制到规则的运动状态, 这种特征对于混沌控制具有重要的利用价值. 上述丰富动力学行为的出现是由于单映像中不连续性和不可逆性相互作用的结果.     

基于时变耦合映像格子模型的信号初值估计

从耦合映像格子中,恢复系统初始条件是耦合系统求逆问题,也是信号处理研究中的一个关键性问题.本文在符号动力学方法的基础上,对映像系数进行修正,针对耦合单峰Logistic映射,提出一种基于时变映像系数恢复信号初值的新方法.在映像过程无噪或受到高斯白噪声污染时,本文方法都能够较好地恢复信号初值的统计特性,而且具有较小的偏差和均方误差,并与原信号之间具有较强的相关性,从而能够更好和更加合理地刻画实际信号的物理过程,对系统初值做出更优的估计. 关键词： 耦合映像格子 恢复初值的统计特性 时变映像系数     

从耦合映象格子中粗略恢复初值的统计特性

耦合映象格子用于信号处理研究时，从中恢复出初始条件是一个非常重要的问题.提出一种粗略恢复格点初值的方法，数值实验表明，动力学函数使用Logistic映射时，在映象过程不存在噪声的情况下，恢复的整个格子初始信号平均值等于给定信号分布的真实平均值，而恢复信号的方差小于给定信号的真实方差.将耦合看作是对独立映射的一种变换，对此作了初步解释，同时发现Logistic映射不同参数下的符号序列排序存在一些有趣的规律.对耦合格子映射研究、非线性耦合则量等是非常有启发意义的. 关键词： 耦合映射格子 信号恢复的统计特性     

Experimental system of coupled map lattices

We design an optical feedback loop system consisting of a liquid-crystal spatial light modulator (SLM), a lens, polarizers, a CCD camera, and a computer. The system images every SLM pixel onto one camera pixel. The light intensity on the camera pixel shows a nonlinear relationship with the phase shift applied by the SLM. Every pixel behaves as a nonlinear map, and we can control the interaction of pixels. Therefore, this feedback loop system can be regarded as a spatially extended system. This experimental coupled map has variable dimensions, which can be up to 512 by 512. The system can be used to study high-dimensional problems that computer simulations cannot handle.     

Partial and complete periodic synchronization in coupled discontinuous map lattices

The partial and complete periodic synchronization in coupled discontinuous map lattices consisting of both discontinuous and non-invertible maps are discussed. We classify three typical types of periodic synchronization states, which give rise to different spatiotemporal patterns including static partial periodic synchronization, dynamically periodic synchronization, and complete periodic synchronization patterns. A special prelude dynamics of partial and complete periodic synchronization motion, which is shown by five separated concave curves in the time series plots of the order parameters,is observed. The detailed analysis shows that the special prelude dynamics is induced by the competition between two synchronized clusters, and the analytical expression for the corresponding order parameter is obtained.     

基于多重耦合映像格子的信号初值恢复研究

利用耦合映像格子恢复信号初值是信号处理研究中一个重要的问题.耦合映像格子具有混沌系统的初值敏感性，当初值受到噪声污染时将会影响到系统对其的恢复.提出了一种由多个一维耦合映像格子系统并列耦合而成的多重耦合映像格子系统，通过将多个一维系统耦合，使因受到噪声干扰而趋向于指数分离的混沌轨道相互靠近，以达到抑制噪声的目的.数值仿真表明，该系统具有较强的抗噪声能力和较高的鲁棒性.在耦合系数选取适当的情况下，即使初始信号受到噪声干扰，该多重耦合系统仍然能够很好地恢复信号初值的统计特性，且对单个初值的恢复情况及与初始信号 关键词： 耦合映像格子 恢复信号的统计特性 多重耦合     

耦合映像格子中时空混沌的状态反馈控制

利用状态反馈的方法，实现了耦合映像格子中时空混沌的稳定控制.反馈方式可以是没有延迟的，也可以是有延迟的；控制方式可以是连续的，也可以是脉冲的.数值模拟结果证明了状态反馈方法的有效性. 关键词： 耦合映像格子 时空混沌 状态反馈控制     

基于单向耦合映象格子生成伪随机位序列的两种新方法

基于时空混沌单向耦合映象格子模型,提出生成伪随机位序列的两种新方法：利用耦合映象格子状态的不变分布特性,选取合适基准对系统中格点状态序列进行判决,生成伪随机位序列;以及基于方向相的思想,通过比较一个格点相邻时间的两个状态值来生成伪随机位序列.对生成序列的性能进行了详细分析,数值实验结果表明,它们具有理想的平衡性、相关性和游程特性,可以应用于信息安全、密码学和数字通信等领域. 关键词： 伪随机位序列 单向耦合映象格子 不变分布 方向相     

Studies of phase return map and symbolic dynamics in a periodically driven Hodgkin–Huxley neuron

How neuronal spike trains encode external information is a hot topic in neurodynamics studies.In this paper,we investigate the dynamical states of the Hodgkin–Huxley neuron under periodic forcing.Depending on the parameters of the stimulus,the neuron exhibits periodic,quasiperiodic and chaotic spike trains.In order to analyze these spike trains quantitatively,we use the phase return map to describe the dynamical behavior on a one-dimensional(1D)map.According to the monotonicity or discontinuous point of the 1D map,the spike trains are transformed into symbolic sequences by implementing a coarse-grained algorithm—symbolic dynamics.Based on the ordering rules of symbolic dynamics,the parameters of the external stimulus can be measured in high resolution with finite length symbolic sequences.A reasonable explanation for why the nervous system can discriminate or cognize the small change of the external signals in a short time is also presented.     

Control of a coupled map lattice model for vortex shedding in the wake of a cylinder

The flow behind a vibrating flexible cable at low Reynolds numbers can exhibit complex wake structures such as lace-like patterns, vortex dislocations and frequency cells. These structures have been observed in experiments and numerical simulations, and are predicted by a previously developed low-order coupled map lattice (CML). The discrete (in time and space) CML models consist of a series of diffusively coupled circle map oscillators along the cable span. Motivated by a desire to modify the complex wake patterns behind flexible vibrating cables we have studied the addition of control terms into the highly efficient CML models and explored the resulting dynamics. Proportional, adaptive proportional and discontinuous non-linear (DNL) control methods were used to derive the control laws. The first method employed occasional proportional feedback. The adaptive method used spatio-temporal feedback control. The DNL method used a discontinuous feedback linearization procedure, and the controller was designed for the resulting linearized system using eigenvalue assignment. These techniques were applied to a modeled vortex dislocation structure in the wake of a vibrating cable in uniform freestream flow. Parallel shedding patterns were achieved for a range of forcing frequency-forcing amplitude combinations studied to validate the control theory. The adaptive proportional and DNL methods were found to be more effective than the proportional control method due to the incorporation of a spatially varying feedback gain across the cylinder span. The DNL method was found to be the most efficient controller of the low-order CML model. The required control level across the cable span was correlated to the 1/1 lock-on behavior of the temporal circle map.     

基于符号动力学的认知事件相关电位的复杂度分析

引入了符号动力学方法分析认知事件相关电位(ERP)的复杂度.以混合模型生成的随机时间序列为例,对近似熵和符号熵作了比较.应用符号熵分析了Oddball范式中不同任务条件(靶刺激和非靶刺激)下的ERP的复杂度.研究发现,额区、中央区和顶区的ERP复杂度在刺激呈现后的任务加工时间段内显著减小(非靶刺激和靶刺激分别在刺激呈现后200—300和400—500ms),而且靶刺激ERP复杂度大约在P300成分的峰值时刻达到最小值,在响应之后逐渐回升.这表明基于符号动力学的复杂度分析能够反映认知任务加工的时间过程,并且 关键词： 事件相关电位 符号动力学 熵     

A control method for congested traffic in the coupled map car-following model

Based on the pioneer work of Konishi et al, a new control method is presented to suppress the traffic congestion in the coupled map (CM) car-following model under an open boundary. A control signal concluding the velocity differences of the two vehicles in front is put forward. The condition under which the traffic jam can be contained is analyzed. The results are compared with that presented by Konishi et al [Phys. Rev. 1999 E 60 4000--4007]. The simulation results show that the temporal behavior obtained by our method is better than that by the Konishi's et al. method, although both the methods could suppress the traffic jam. The simulation results are consistent with the theoretical analysis.     

混沌伪随机序列复杂度分析的符号动力学方法

通过将混沌伪随机序列看成一个符号序列，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.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，该方法可以有效地判断出不同的混沌伪随机序列的复杂程度，而且比近似熵法更为 优越. 关键词： 混沌 伪随机序列 符号动力学 熵     

Feedback control scheme of traffic jams based on the coupled map car-following model

Based on the pioneering work of Konishi et al. [Phys. Rev. E （1999） 60 4000], a new feedback control scheme is presented to suppress traffic jams based on the coupled map car-following model under the open boundary condition. The effect of the safe headway on the traffic system is considered. According to the control theory, the condition under which traffic jams can be suppressed is analyzed. The results are compared with the previous results concerning congestion control. The simulations show that the suppression performance of our scheme on traffic jams is better than those of the previous schemes, although all the schemes can suppress traffic jams. The simulation results are consistent with theoretical analyses.     

Fractal patterns on the onset of coherent structures in a coupled map lattice

We report the formation of Cantor set-like fractals during the development of coherent structures in a coupled map lattice (CML). The dependence of these structures on the size of the lattice as well as the first three dimensions of the associated fractal patterns are analyzed numerically.