Recent development of chaos theory in topological dynamics

We give a summary on the recent development of chaos theory in topological dynamics,focusing on Li–Yorke chaos, Devaney chaos, distributional chaos, positive topological entropy, weakly mixing sets and so on, and their relationships.     

多级混沌映射变参数伪随机序列产生方法研究

针对单混沌系统因计算机有限精度效应产生的混沌退化问题,提出了一种多级混沌映射变参数伪随机序列产生方法,基于该方法构建的混沌系统较单混沌系统具有伪随机序列周期大,密钥数量多,密钥空间大等优势,所产生的密码具有更高的安全性能.仿真结果表明,该方法在低复杂度条件下可以生成大量具有良好自相关和互相关特性的混沌序列,在安全领域具有良好的应用前景.     

A Devaney Chaotic System Which Is Neither Distributively nor Topologically Chaotic

Weiss proved that Devaney chaos does not imply topological chaos and Oprocha pointed out that Devaney chaos does not imply distributional chaos. In this paper, by constructing a simple example which is Devaney chaotic but neither distributively nor topologically chaotic, we give a unified proof for the results of Weiss and Oprocha.     

A note on distributional chaos with respect to a sequence

The aim of this note is to use methods developed by Kuratowski and Mycielski to prove that some more common notions in topological dynamics imply distributional chaos with respect to a sequence. In particular, we show that the notion of distributional chaos with respect to a sequence is only slightly stronger than the definition of chaos due to Li and Yorke. Namely, positive topological entropy and weak mixing both imply distributional chaos with respect to a sequence, which is not the case for distributional chaos as introduced by Schweizer and Smítal.     

Distributional chaos for backward shifts

We provide sufficient conditions which give uniform distributional chaos for backward shift operators. We also compare distributional chaos with other well-studied notions of chaos for linear operators, like Devaney chaos and hypercyclicity, and show that Devaney chaos implies uniform distributional chaos for weighted backward shifts, but there are examples of backward shifts which are uniformly distributionally chaotic and not hypercyclic.     

Recurrence and symmetry of time series: Application to transition detection

The study of transitions in low dimensional, nonlinear dynamical systems is a complex problem for which there is not yet a simple, global numerical method able to detect chaos–chaos, chaos–periodic bifurcations and symmetry-breaking, symmetry-increasing bifurcations. We present here for the first time a general framework focusing on the symmetry concept of time series that at the same time reveals new kinds of recurrence. We propose several numerical tools based on the symmetry concept allowing both the qualification and quantification of different kinds of possible symmetry. By using several examples based on periodic symmetrical time series and on logistic and cubic maps, we show that it is possible with simple numerical tools to detect a large number of bifurcations of chaos–chaos, chaos–periodic, broken symmetry and increased symmetry types.     

Chaos for Subshifts of Finite Type

This paper deals with chaos for subshifts of finite type. We show that for any subshift of finite type determined by an irreducible and aperiodic matrix, there is a finitely chaotic set with full Hausdorff dimension. Moreover, for any subshift of finite type determined by a matrix, we point out that the cases including positive topological entropy, distributional chaos, chaos and Devaney chaos are mutually equivalent.     

Chaos and Shadowing Around a Heteroclinically Tubular Cycle With an Application to Sine-Gordon Equation

The theory of chaos and shadowing developed recently by the author is amplified to the case of a heteroclinically tubular cycle. Specifically, let F be a   C ³  diffeomorphism on a Banach space. F has a heteroclinically tubular cycle that connects two normally hyperbolic invariant manifolds. Around the heteroclinically tubular cycle, a Bernoulli shift dynamics of submanifolds is established through a shadowing lemma. As an example, a sine-Gordon equation with a chaotic forcing is studied. Existence of a heteroclinically tubular cycle is proved. Also proved are chaos associated with the heteroclinically tubular cycle, and chaos cascade referring to the embeddings of smaller-scale chaos in larger-scale chaos.     

Statistical properties of quantum spectra in nuclei

Some aspects of quantum chaos in a finite system have been studied based on the analysis of statistical behavior of quantum spectra in nuclei. The experiment data show the transition from order to chaos with increasing excitation energy in spherical nuclei. The dependence of the order to chaos transition on nuclear deformation and nuclear rotating is described. The influence of pairing effect on the order to chaos transition is also discussed. Some important experiment phenomena in nuclear physics have been understood from the point of view of the interplay between order and chaos.     

Anti-control of chaos of single time scale brushless dc motors and chaos synchronization of different order systems

Anti-control of chaos of single time scale brushless dc motors (BLDCM) and chaos synchronization of different order systems are studied in this paper. By addition of an external nonlinear term, we can obtain anti-control of chaos. Then, by addition of the coupling terms, by the use of Lyapunov stability theorem and by the linearization of the error dynamics, chaos synchronization between a third-order BLDCM and a second-order Duffing system are presented.     

Martelli’s Chaos in Inverse Limit Dynamical Systems and Hyperspace Dynamical Systems

In this paper we investigate Martelli’s chaos of inverse limit dynamical systems and hyperspace dynamical systems which are both induced from dynamical systems on a compact metric space. We give the implication of Martelli’s chaos among those systems. More precisely, we show that inverse limit dynamical system is Martelli’s chaos if and only if so is original system, and we prove that hyperspace dynamical system is Martelli’s chaos implies original system is Martelli’s chaos if the orbit of every single point set of original system is unstable in hyperspace dynamical system.     

High level chaos in the exchange and index markets

Many studies were inconclusive about the presence of chaos in financial markets due to test misspecification. Chaos tests present in the literature need noise-free time series, since any measurement error will induce the rejection of chaos. Moreover, chaos was merely tested on a low-level basis. This paper investigates the presence of a high-level noisy chaos in financial data; simulations were conclusive about the power of the test. When applied to six stock indexes and six exchange rates, the hypothesis of chaotic dynamics was rejected for all data.     

Li-Yorke and distributionally chaotic operators

We study Li-Yorke chaos and distributional chaos for operators on Banach spaces. More precisely, we characterize Li-Yorke chaos in terms of the existence of irregular vectors. Sufficient “computable” criteria for distributional and Li-Yorke chaos are given, together with the existence of dense scrambled sets under some additional conditions. We also obtain certain spectral properties. Finally, we show that every infinite dimensional separable Banach space admits a distributionally chaotic operator which is also hypercyclic.     

Distributional chaos in random dynamical systems

In this paper, we introduce the notion of distributional chaos and the measure of chaos for random dynamical systems generated by two interval maps. We give some sufficient conditions for a zero measure of chaos and examples of chaotic systems. We demonstrate that the chaoticity of the functions that generate a system does not, in general, affect the chaoticity of the system, i.e. a chaotic system can arise from two nonchaotic functions and vice versa. Finally, we show that distributional chaos for random dynamical system is, in some sense, unstable.     

混沌在水质不确定性预测中的应用

在Dobbins-Camp模型的基础上,利用混沌理论分析了水质预测中的误差,得到关于误差时间序列的相空间,在此基础上,将混沌神经网络嵌入Dobbins-Camp模型,建立一个具有混沌特性和学习功能的水质不确定性模型,仿真结果表明,方法在预测水质不确定性上是有效的.     

Quasi-cycles and sensitive dependence on seed values in edge of chaos behaviour in a class of self-evolving maps

In a recent paper [Melby P, Kaidel J, Weber N, Hubler A. Adaptation to the edge of chaos in the self-adjusting logistic map. Phys Rev Lett 2000;84:5991–3], Melby et al. attempted to understand edge of chaos behaviour through a very simple model. Based on our exhaustive numerical experiments, here we show that the model, with the definition of the edge of chaos given in the paper, cannot unequivocally support the idea of adaptation to the edge of chaos, let alone allow a conjecture of its generic presence in systems having the same characteristic features.     

Coherence properties of cycling chaos

Cycling chaos is a heteroclinic connection between several chaotic attractors, at which switchings between the chaotic sets occur at growing time intervals. Here we characterize the coherence properties of these switchings, considering nearly periodic regimes that appear close to the cycling chaos due to imperfections or to instability. Using numerical simulations of coupled Lorenz, Roessler, and logistic map models, we show that the coherence is high in the case of imperfection (so that asymptotically the cycling chaos is very regular), while it is low close to instability of the cycling chaos.     

Bifurcations and chaos in Mira 2 map

In this paper, Mira 2 map is investigated. The conditions of the existence for fold bifurcation, flip bifurcation and Naimark-Sacker bifurcation are derived by using center manifold theorem and bifurcation theory. And the conditions of the existence for chaos in the sense of Marroto are obtained. Numerical simulation results not only show the consistence with the theoretical analysis but also display complex dynamical behaviors, including period-n orbits, crisis, some chaotic attractors, period-doubling bifurcation to chaos, quasi-period behaviors to chaos, chaos to quasi-period behaviors, bubble and onset of chaos.