A method of recovering the initial vectors of globally coupled map lattices based on symbolic dynamics

Based on symbolic dynamics, a novel computationally efficient algorithm is proposed to estimate the unknown initial vectors of globally coupled map lattices (CMLs). It is proved that not all inverse chaotic mapping functions are satisfied for contraction mapping. It is found that the values in phase space do not always converge on their initial values with respect to sufficient backward iteration of the symbolic vectors in terms of global convergence or divergence (CD). Both CD property and the coupling strength are directly related to the mapping function of the existing CML. Furthermore, the CD properties of Logistic, Bernoulli, and Tent chaotic mapping functions are investigated and compared. Various simulation results and the performances of the initial vector estimation with different signal-to-noise ratios (SNRs) are also provided to confirm the proposed algorithm. Finally, based on the spatiotemporal chaotic characteristics of the CML, the conditions of estimating the initial vectors using symbolic dynamics are discussed. The presented method provides both theoretical and experimental results for better understanding and characterizing the behaviours of spatiotemporal chaotic systems.     

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).     

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.     

Cascades with coupled map lattices in preferential attachment community networks

In this paper, cascading failure is studied by coupled map lattice （CML） methods in preferential attachment community networks. It is found that external perturbation R is increasing with modularity Q growing by simulation. In particular, the large modularity Q can hold off the cascading failure dynamic process in community networks. Furthermore, different attack strategies also greatly affect the cascading failure dynamic process. It is particularly significant to control cascading failure process in real community networks.     

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     

The emergence of coherent structures in coupled map lattices

We show how increasing spatial interaction leads to the merging of coherent structures from chaos in some systems of coupled map lattices. This phenomenon reflects the arising of new ground states in the corresponding model of statistical mechanics. If we further increase the coupling then, new ground states appear showing the coexistence of a large-scale coherent structure with a small-scale chaotic motion. This allows us to propose a generalization of the notion of spatial intermittency.     

Existence and stability of steady fronts in bistable coupled map lattices

We prove the existence and we study the stability of the kinklike fixed points in a simple coupled map lattice (CML) for which the local dynamics has two stable fixed points. The condition for the existence allows us to define a critical value of the coupling parameter where a (multi) generalized saddle-node bifurcation occurs and destroys these solutions. An extension of the results to other CMLs in the same class is also displayed. Finally, we emphasize the property of spatial chaos for small coupling.     

Tolerance of edge cascades with coupled map lattices methods

This paper studies the cascading failure on random networks and scale-free networks by introducing the tolerance parameter of edge based on the coupled map lattices methods. The whole work focuses on investigating some indices including the number of failed edges, dynamic edge tolerance capacity and the perturbation of edge. In general, it assumes that the perturbation is attributed to the normal distribution in adopted simulations. By investigating the effectiveness of edge tolerance in scale-free and random networks, it finds that the larger tolerance parameter λ can more efficiently delay the cascading failure process for scale-free networks than random networks. These results indicate that the cascading failure process can be effectively controlled by increasing the tolerance parameter λ. Moreover, the simulations also show that, larger variance of perturbation can easily trigger the cascading failures than the smaller one. This study may be useful for evaluating efficiency of whole traffic systems, and for alleviating cascading failure in such systems.     

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.     

Symbolic dynamics of two coupled Lorenz maps: From uncoupled regime to synchronisation

The bounded dynamics of a system of two coupled piecewise affine and chaotic Lorenz maps is studied over the coupling range, from the uncoupled regime where the entropy is maximal, to the synchronized regime where the entropy is minimal. By formulating the problem in terms of symbolic dynamics, bounds on the set of orbit codes (or the set itself, depending on parameters) are determined which describe the way the dynamics is gradually affected as the coupling increases. Proofs rely on monotonicity properties of bounded orbit coordinates with respect to some partial ordering on the corresponding codes. The estimates are translated in terms of (bounds on the) entropy, which are monotonically decreasing with coupling and which are compared to the numerically computed entropy. A good agreement is found which indicates that these bounds capture the essential features of the transition from the uncoupled regime to synchronisation.     

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.     

Influence of thermalization on the initial condition for heavy ion collisions

In this paper, we discuss the important role of the thermalization process in the initial distribution of QGP. We find that the negligible heat conduction inside QGP can be expressed as an effective Fourier law and we further analyse qualitatively the results caused by a thermalized initial condition. Based on this arguments, we construct a simple phenomenological model and work with the hydro code, and then we compare our results with the experimental data and the results of the standard initial model. It is found that, as we have argued, a thermalized initial condition suppresses the value of the elliptic flow.     

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.     

Distribution of periodic orbits for the Casati-Prosen map on rational lattices

We study numerically the periodic orbits of the Casati-Prosen map, a two-parameter reversible map of the torus, with zero entropy. For rational parameter values, this map preserves rational lattices, and each lattice decomposes into periodic orbits. We consider the distribution function of the periods over prime lattices, and its dependence on the parameters of the map. Based on extensive numerical evidence, we conjecture that, asymptotically, almost all orbits are symmetric, and that for a set of rational parameters having full density, the distribution function approaches the gamma-distribution R(x)=1−e^−x(1+x). These properties, which have been proved to hold for random reversible maps, were previously thought to require a stronger form of deterministic randomness, such as that displayed by rational automorphisms over finite fields. Furthermore, we show that the gamma-distribution is the limit of a sequence of singular distributions which are observed on certain lines in parameter space. Our experiments reveal that the convergence rate to R is highly non-uniform in parameter space, being slowest in sharply-defined regions reminiscent of resonant zones in Hamiltonian perturbation theory.     

Exact treatment of mode locking for a piecewise linear map

A piecewise linear map with one discontinuity is studied by analytic means in the two-dimensional parameter space. When the slope of the map is less than unity, periodic orbits are present, and we give the precise symbolic dynamic classification of these. The localization of the periodic domains in parameter space is given by closed expressions. The winding number forms a devil's terrace, a two-dimensional function whose cross sections are complete devils's staircases. In such a cross section the complementary set to the periodic intervals is a Cantor set with dimensionD=0.     

The study of banded spherulite patterns by coupled logistic map lattice model

Banded spherulite patterns are simulated in two dimensions by means of a coupled logistic map lattice model. Both target pattern and spiral pattern which have been proved to be existent experimentally in banded spherulite are obtained by choosing suitable parameters in the model. The simulation results also indicate that the band spacing is decreased with the increase of parameter μ in the logistic map and increased with the increase of the coupling parameter ε, which is quite similar to the results in some experiments. Moreover, the relationship between the parameters and the corresponding patterns is obtained, and the target patterns and spiral patterns are distinguished for a given group of initial values, which may guide the study of banded spherulite.     

Security analysis of a multiple pseudorandom-bit generator based on a spatiotemporal chaotic map

<正>In this paper we analyse the security of a multiple pseudorandom-bit generator based on the coupled map lattice and suggest an improved model.Utilizing the error function attack,the multiple pseudorandom-bit generators which can be realized by the three digitization methods are analysed and the effective key spaces are estimated.We suggest an improved multiple pseudorandom-bit generator with 128-bit secret key,and analyse the key sensitivity and statistical properties of the system.     

On the rate of convergence to the normal law for solutions of the Burgers equation with singular initial data

We study the scaling limit of random fields which are the solutions of a nonlinear partial differential equation known as the Burgers equation, under stochastic initial condition. These are assumed to be a Gaussian process with long-range dependence. We present some results on the rate of convergence to the normal law.     

Analysis and engineering of coupled cavity waveguides based on coupled-mode theory

The analytical expression for the transmission spectra of coupled cavity waveguides （CCWs） in photonic crystals （PCs） is derived based on the coupled-mode theory （CMT）. Parameters in the analytical expression can be extracted by simple numerical simulations. We reveal that it is the phase shift between the two adjacent PC defects that uniquely determines the flatness of the impurity bands of CCWs. In addition, it is found that the phase shift also greatly affects the bandwidth of CCWs. Thus, the engineering of the impurity bands of CCWs can be realized through the adjustment of the phase shift. Based on the theoretical results, an interesting phenomenon in which a CCW acts as a single PC defect and its impurity band possesses a Lorentz lineshape is predicted. Very good agreement between the analytical results and the numerical simulations based on transfer matrix method has been achieved.