Topological invariants in the study of a chaotic food chain system

The study of ecological systems has generated deep interest in exploring the complexity of chaotic food chains. The role of chaos in ecosystems is not entirely understood. One approach to have a better comprehension of ecological chaos is by analyzing it in mathematical models of basic food chains. In this article it is considered a classical chaotic food chain model from the literature. We use the theory of symbolic dynamics to study the topological entropy and the parameter space ordering of kneading sequences associated with one-dimensional maps that reproduce significant aspects of the model dynamics. The topological entropy allows us to distinguish different chaotic states in some realistic system parameter region. Another numerical invariant is introduced in order to characterize isentropic dynamics. Studying a set of maps with the same topological entropy, we exhibit numerical results about the relation between the second topological invariant and each of the control parameters in consideration. This work provides an illustration of how our understanding of ecological models can be enhanced by the theory of symbolic dynamics.     

Validity of threshold-crossing analysis of symbolic dynamics from chaotic time series

A practical and popular technique to extract the symbolic dynamics from experimentally measured chaotic time series is the threshold-crossing method, by which an arbitrary partition is utilized for determining the symbols. We address to what extent the symbolic dynamics so obtained can faithfully represent the phase-space dynamics. Our principal result is that such a practice can lead to a severe misrepresentation of the dynamical system. The measured topological entropy is a Devil's staircase-like, but surprisingly nonmonotone, function of a parameter characterizing the amount of misplacement of the partition.     

Quantifying chaos for ecological stoichiometry

The theory of ecological stoichiometry considers ecological interactions among species with different chemical compositions. Both experimental and theoretical investigations have shown the importance of species composition in the outcome of the population dynamics. A recent study of a theoretical three-species food chain model considering stoichiometry [B. Deng and I. Loladze, Chaos 17, 033108 (2007)] shows that coexistence between two consumers predating on the same prey is possible via chaos. In this work we study the topological and dynamical measures of the chaotic attractors found in such a model under ecological relevant parameters. By using the theory of symbolic dynamics, we first compute the topological entropy associated with unimodal Poincare? return maps obtained by Deng and Loladze from a dimension reduction. With this measure we numerically prove chaotic competitive coexistence, which is characterized by positive topological entropy and positive Lyapunov exponents, achieved when the first predator reduces its maximum growth rate, as happens at increasing δ1. However, for higher values of δ1 the dynamics become again stable due to an asymmetric bubble-like bifurcation scenario. We also show that a decrease in the efficiency of the predator sensitive to prey's quality (increasing parameter ζ) stabilizes the dynamics. Finally, we estimate the fractal dimension of the chaotic attractors for the stoichiometric ecological model.     

Symbolic Dynamics and Topological Entropy of Henon Map

By using the method with which the trajectories of dissipative maps can be derived from the Hamiltonian, we study the symbolic dynamics of Henon map and its relation with the symbolic dynamics of unimodal map, and compute the topological entropy as a function of the parameter a and b. Finally, the boundary of the region where the topological entropy exists on the parameter plane is given.     

混沌信号自适应协同滤波去噪

混沌信号协同滤波去噪算法充分利用了混沌信号的自相似结构特征,具有良好的信噪比提升性能.针对该算法的滤波参数优化问题,考虑到最优滤波参数的选取受到信号特征、采样频率和噪声水平的影响,为提高该算法的自适应性使其更符合实际应用需求,基于排列熵提出一种滤波参数自动优化准则.依据不同噪声水平的混沌信号排列熵的不同,首先选取不同滤波参数对含噪混沌信号进行去噪,然后计算各滤波参数对应重构信号的排列熵,最后通过比较各重构信号的排列熵,选取排列熵最小的重构信号对应的滤波参数为最优滤波参数,实现滤波参数的优化.分析了不同信号特征、采样频率和噪声水平情况下滤波参数的选取规律.仿真结果表明,该参数优化准则能在不同条件下对滤波参数进行有效的自动最优化,提高了混沌信号协同滤波去噪算法的自适应性.     

Symbolic dynamics of one-dimensional maps: Entropies,finite precision,and noise

In the study of nonlinear physical systems, one encounters apparently random or chaotic behavior, although the systems may be completely deterministic. Applying techniques from symbolic dynamics to maps of the interval, we compute two measures of chaotic behavior commonly employed in dynamical systems theory: the topological and metric entropies. For the quadratic logistic equation, we find that the metric entropy converges very slowly in comparison to maps which are strictly hyperbolic. The effects of finite precision arithmetric and external noise on chaotic behavior are characterized with the symbolic dynamics entropies. Finally, we discuss the relationship of these measures of chaos to algorithmic complexity, and use algorithmic information theory as a framework to discuss the construction of models for chaotic dynamics.     

用符号分析法显示时空混沌信号之间的相互关系

基于耦合映射格子模型, 用符号分析法研究了时空混沌信号的相互关系. 数值计算结果表明, 两个直接耦合的格子之间或两个未直接耦合但邻近的格子之间的条件熵存在一个尖锐的极小值, 而两个距离比较远的格子之间的条件熵则没有尖锐的极小值. 因此, 耦合映射格子系统产生的时空混沌信号之间的相互关系可以通过符号分析的方法显示出来.     

Estimating the Shannon entropy: recurrence plots versus symbolic dynamics

Recurrence plots were first introduced to quantify the recurrence properties of chaotic dynamics. A few years later, the recurrence quantification analysis was introduced to transform graphical representations into statistical analysis. Among the different measures introduced, a Shannon entropy was found to be correlated with the inverse of the largest Lyapunov exponent. The discrepancy between this and the usual interpretation of a Shannon entropy is solved here by using a new definition--still based on the recurrence plots--and it is verified that this new definition is correlated with the largest Lyapunov exponent, as expected from the Pesin conjecture. A comparison with a Shannon entropy computed from symbolic dynamics is also provided.     

Characterizing the dynamics of coupled pendulums via symbolic time series analysis

We propose a novel method of symbolic time-series analysis aimed at characterizing the regular or chaotic dynamics of coupled oscillators. The method is applied to two identical pendulums mounted on a frictionless platform, resembling Huygens’ clocks. Employing a transformation rule inspired in ordinal analysis [C. Bandt and B. Pompe, Phys. Rev. Lett. 88, 174102 (2002)], the dynamics of the coupled system is represented by a sequence of symbols that are determined by the order in which the trajectory of each pendulum intersects an appropriately chosen hyperplane in the phase space. For two coupled pendulums we use four symbols corresponding to the crossings of the vertical axis (at the bottom equilibrium point), either clock-wise or anti-clock wise. The complexity of the motion, quantified in terms of the entropy of the symbolic sequence, is compared with the degree of chaos, quantified in terms of the largest Lyapunov exponent. We demonstrate that the symbolic entropy sheds light into the large variety of different periodic and chaotic motions, with different types synchronization, that cannot be inferred from the Lyapunov analysis.     

Complex dynamics of cellular automata rule 119

In this paper, the dynamical behaviors of cellular automata rule 119 are studied from the viewpoint of symbolic dynamics in the bi-infinite symbolic sequence space Σ₂. It is shown that there exists one Bernoulli-measure global attractor of rule 119, which is also the nonwandering set of the rule. Moreover, it is demonstrated that rule 119 is topologically mixing on the global attractor and possesses the positive topological entropy. Therefore, rule 119 is chaotic in the sense of both Li-Yorke and Devaney on the global attractor. It is interesting that rule 119, a member of Wolfram’s class II which was said to be simple as periodic before, actually possesses a chaotic global attractor in Σ₂. Finally, it is noted that the method presented in this work is also applicable to studying the dynamics of other rules, especially the 112 Bernoulli-shift rules therein.     

一维简并双参数四次方映射的符号动力学

研究了一维简并双参数四次方映射的符号动力学，构造了揉平面的超稳轨道的骨架图及关节点。根据拓扑熵定义，给出了拓扑熵混沌存在的判据。 关键词：     

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.     

The Generalized Milnor–Thurston Conjecture and Equal Topological Entropy Class in Symbolic Dynamics of Order Topological Space of Three Letters

This paper presents an answer to an open problem in the dynamical systems of three letters: the generalized Milnor–Thurston conjecture on the existence of infinitely many plateaus of topological entropy in the two-dimensional parameter plane. The concept of equal topological entropy class is introduced by the dual star product which is a generalization of the Derrida–Gervois–Pomeau star product to the symbolic dynamics of three letters for the endomorphisms on the interval. The algebraic rules established by the dual star products for the doubly superstable kneading sequences are equivalent to the normal factorization of the Milnor–Thurston characteristic polynomials. Moreover, the classification theory of symbolic primitive and compound sequences based on the topological conjugacy in the meaning of equal entropy is completed in the topological space Σ₃ of three letters. Received: 4 February 1998 / Accepted: 1 March 2000     

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.     

Globally enumerating unstable periodic orbits for observed data using symbolic dynamics

The unstable periodic orbits of a chaotic system provide an important skeleton of the dynamics in a chaotic system, but they can be difficult to find from an observed time series. We present a global method for finding periodic orbits based on their symbolic dynamics, which is made possible by several recent methods to find good partitions for symbolic dynamics from observed time series. The symbolic dynamics are approximated by a Markov chain estimated from the sequence using information-theoretical concepts. The chain has a probabilistic graph representation, and the cycles of the graph may be exhaustively enumerated with a classical deterministic algorithm, providing a global, comprehensive list of symbolic names for its periodic orbits. Once the symbolic codes of the periodic orbits are found, the partition is used to localize the orbits back in the original state space. Using the periodic orbits found, we can estimate several quantities of the attractor such as the Lyapunov exponent and topological entropy.     

一种确定混沌伪随机序列复杂度的模糊关系熵测度

将模糊关系的概念引入混沌伪随机序列复杂度的测度方法之中,提出了一种新的混沌伪随机序列复杂度测度方法——模糊关系熵(fuzzy relationship entropy,简记为F-REn)测度方法,并推导了F-REn的两个基本性质.仿真结果表明,该测度方法能够有效测度混沌伪随机序列的复杂度,与近似熵(ApEn)测度方法和符号熵测度方法相比,F-REn测度具有更加好的对序列符号空间的适用性、更加小的对测量维度的敏感性和更加强的对分辨率参数的鲁棒性. 关键词： 混沌伪随机序列 模糊理论 复杂度     

Symbolic dynamics and topological entropy at the onset of pruning

The topological entropy and pruning rules are investigated for two-dimensional smooth maps at the onset of pruning. Typically the difference of the parameter-dependent topological entropy from its maximum value increases with a power law. Superimposed on this decrease, there are periodic or quasiperiodic oscillations on a logarithmic scale. Both, the scaling exponent and the periodicity are determined by the Lyapunov exponents of the first pruned orbit and the minimal number of letters in the alphabet of the symbolic dynamics. If, at the onset of pruning, the averaged Lyapunov exponent is sufficiently large and the first pruned orbit is homoclinic, the entropy function of area-preserving maps exhibits a series of plateaux. On the plateaux, the symbolic dynamics can be described by finitely many finite forbidden words. There is a series of plateaux which, in different systems, can be described by the same type of forbidden words.     

Chaotic scattering: An introduction

In recent years chaotic behavior in scattering problems has been found to be important in a host of physical situations. Concurrently, a fundamental understanding of the dynamics in these situations has been developed, and such issues as symbolic dynamics, fractal dimension, entropy, and bifurcations have been studied. The quantum manifestations of classical chaotic scattering is also an extremely active field, with new analytical techniques being developed and with experiments being carried out. This issue of Chaos provides an up-to-date survey of the range of work in this important field of study.     

Dynamical quantities and their numerical analysis by saddle periodic orbits

We study estimators for dynamical quantities such as the topological entropy and the topological pressure which are based on numerical computations on periodic orbits. For the particular case of the Hénon family for some parameter ranges we find a reasonable convergence of the entropy, the pressure, and Birkhoff averages of test functions. However, pointing out possible pitfalls of such an analysis, we show that the evaluation by means of saddle orbits alone can be misleading if, for example, chaotic saddles and attractors coexist.