Symbolic Dynamics of the Forced Brusselator

The symbolic dynamics of the periodically forced Brusselator for certain parameters is established, based on the binary partition determined from primary tangencies between forward and backward foliations of the Poincaré map. The ordering rule for forward and backward sequences is obtained, and the admissibility condition for allowed sequences is discussed. Some allowed periodic orbits are determined and numerically verified. A method to construct allowed chaotic sequences is proposed.     

Symbolic Dynamics of the Piecewise Linear Standard Map

The symbolic dynamics of the sawtooth map is extended to the dissipative piecewise linear standard map. The ordering rules of foliations are derived, and the admissibility condition for allowed symbolic sequences is obtained. A symbolic analysis is given for invariant circles of the piecewise linear standard map without dissipation.     

Determination of Partition Lines from Dynamical Foliations for the Hénon Map

The binary partition lines for the Hhnon map are numerically constructed from tangencies between the contracting and expanding foliations. The ordering of foliations acording to their symbolic sequences are examined.     

Symbolic Analysis of the Hénon Map at a = 1.4 and b = 0.3

The symbolic dyhamics of the Lozi map is applied to the Hgnon map at a = 1.4 and b = 0.3. The fundamental forbidden zone, inside which points correspond to forbidden symbolic sequences, is constructed from a numerically determined bartition line. All the allowed periodic orbits up to period 7 are determined, and chaotic orbits are derived.     

Symbolic Dynamics for the Tél Map

In terms of contracting and expanding foliations the two-dimensional Tél map is decomposed into two coupled one-dimensional maps. Symbolic sequences are assigned to the two classes of foliations, and their ordering is discussed. The pruning front is constructed. A necessary and sufficient condition for admissible sequences is proposed.     

Symbolic Dynamics for the General Quartic Map

The general quartic map is studied by symbolic dynamics in the three parameter space, joints are obtained in terms of the kneading theory. We describe the skeleton of superstable. kneading sequences corresponding to a joint and all allowed superstable kneading sequences in the kneading space in details. A criterion for the existence of chaos has been given according to the topological entropy.     

How does a choice of Markov partition affect the resultant symbolic dynamics?

The mutual relationship among Markov partitions is investigated for one-dimensional piecewise monotonic map. It is shown that if a Markov partition is regarded as a map-refinement of the other Markov partition, that is, a concept we newly introduce in this article, one can uniquely translate a set of symbolic sequences by one Markov partition to those by the other or vice versa. However, the set of symbolic sequences constructed using Markov partitions is not necessarily translated with each other if there exists no map-refinement relation among them. By using a roof map we demonstrate how the resultant symbolic sequences depend on the choice of Markov partitions.     

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.     

Fairing of Legs in the Parameter Plane of the Hénon Map

The regular structure-parallel to-structure sequence of shrimp shaped isoperiodic domains in the parameter plane of the HJnon map is explained by symbolic dynamics. Rules for the pairing of symbolic sequences and for their ordering in the parameter plane are found.     

Estimation of chaotic coupled map lattices using symbolic vector dynamics

In [K. Wang, W.J. Pei, Z.Y. He, Y.M. Cheung, Phys. Lett. A 367 (2007) 316], an original symbolic vector dynamics based method has been proposed for initial condition estimation in additive white Gaussian noisy environment. The estimation precision of this estimation method is determined by symbolic errors of the symbolic vector sequence gotten by symbolizing the received signal. This Letter further develops the symbolic vector dynamical estimation method. We correct symbolic errors with backward vector and the estimated values by using different symbols, and thus the estimation precision can be improved. Both theoretical and experimental results show that this algorithm enables us to recover initial condition of coupled map lattice exactly in both noisy and noise free cases. Therefore, we provide novel analytical techniques for understanding turbulences in coupled map lattice.     

圆映象的符号动力学

用符号动力学的方法对圆映象的Farey序列、Farey序列与M.S.S.序列的＊积、二元树及各Farey数的新生轨道进行了研究。 关键词：     

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     

A complexity measure approach based on forbidden patterns and correlation degree

Based on forbidden patterns in symbolic dynamics, symbolic subsequences are classified and relations between forbidden patterns, correlation dimensions and complexity measures are studied. A complexity measure approach is proposed in order to separate deterministic (usually chaotic) series from random ones and measure the complexities of different dynamic systems. The complexity is related to the correlation dimensions, and the algorithm is simple and suitable for time series with noise. In the paper, the complexity measure method is used to study dynamic systems of the Logistic map and the H\'enon map with multi-parameters.     

Knot Theory Based on the Minimal Braid in Lorenz System

By means of symbolic dynamics in Lorenz map, after studying spatial topological structure of dynamical knot constructed by the minimal braid assumption, we pry into the spatial structure of three-dimensional manifold from low-dimensional space. Lorenz dynamical knot provides a scheme about suspension. So, we are able to understand partly dynamical behaviors’ topological properties of high-dimensional differential manifold by studying dynamical knot’s properties. We hope to afford an approach and understand the nature of physical reality, especially in the study of DNA sequences, 20 amino acids symbolic sequences of proteid structure, and time series that can be symbolic in finance market et al.     

A circle map in a human heart

A circle is divided into two regions, a black one and a white one. Successive iterates of an invertible nonlinear circle map generate a symbolic string indicating whether each iterate is in the black or white region. A number of remarkable properties of the symbolic sequences are described. These properties were previously described for a linear circle map corresponding to a rigid rotation in the “gaps and steps” problem. These results have direct application to a cardiac arrhythmia, parasystole, that results from the competition between two pacemakers in the heart, one in the sinus mode and the other in the ventricles. The theoretical results are directly applicable to a clinical case of a young man who had frequent extra heartbeats.     

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.     

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