Probability density functions of some skew tent maps

We consider a family of chaotic skew tent maps. The skew tent map is a two-parameter, piecewise-linear, weakly-unimodal, map of the interval F_a,b. We show that F_a,b is Markov for a dense set of parameters in the chaotic region, and we exactly find the probability density function (pdf), for any of these maps. It is well known (Boyarsky A, Góra P. Laws of chaos: invariant measures and dynamical systems in one dimension. Boston: Birkhauser, 1997), that when a sequence of transformations has a uniform limit F, and the corresponding sequence of invariant pdfs has a weak limit, then that invariant pdf must be F invariant. However, we show in the case of a family of skew tent maps that not only does a suitable sequence of convergent sequence exist, but they can be constructed entirely within the family of skew tent maps. Furthermore, such a sequence can be found amongst the set of Markov transformations, for which pdfs are easily and exactly calculated. We then apply these results to exactly integrate Lyapunov exponents.     

一般三角帐篷映射混沌性与两种混沌互不蕴含性

将三角帐篷映射推广为一般的n-三角帐篷映射,并且借助于一般Bernoulli移位映射,Banks定理与Li-Yorke定理,首先证明：对于任意的正整数n,n-三角帐篷映射既是Devaney混沌的,也是Li-Yorke混沌的.然后,利用所得到的结果,通过实例展示：Devaney混沌与Li-Yorke混沌的互不蕴含性.     

Chaos optimization algorithms based on chaotic maps with different probability distribution and search speed for global optimization

Chaos optimization algorithms (COAs) usually utilize the chaotic map like Logistic map to generate the pseudo-random numbers mapped as the design variables for global optimization. Many existing researches indicated that COA can more easily escape from the local minima than classical stochastic optimization algorithms. This paper reveals the inherent mechanism of high efficiency and superior performance of COA, from a new perspective of both the probability distribution property and search speed of chaotic sequences generated by different chaotic maps. The statistical property and search speed of chaotic sequences are represented by the probability density function (PDF) and the Lyapunov exponent, respectively. Meanwhile, the computational performances of hybrid chaos-BFGS algorithms based on eight one-dimensional chaotic maps with different PDF and Lyapunov exponents are compared, in which BFGS is a quasi-Newton method for local optimization. Moreover, several multimodal benchmark examples illustrate that, the probability distribution property and search speed of chaotic sequences from different chaotic maps significantly affect the global searching capability and optimization efficiency of COA. To achieve the high efficiency of COA, it is recommended to adopt the appropriate chaotic map generating the desired chaotic sequences with uniform or nearly uniform probability distribution and large Lyapunov exponent.     

Chaos in the delay logistic equation with discontinuous delays

This paper analyzes a delay logistic equation which models a feedback control problem. Interval map associated to the system is derived. By calculating Lyapunov exponent, we indicate stable orbit and chaotic phenomenon respectively. The results are verified through computer simulation. We identify the parameter which controls the dynamics.     

Robust chaos in a credit cycle model defined by a one-dimensional piecewise smooth map

We consider a family of one-dimensional continuous piecewise smooth maps with monotone increasing and monotone decreasing branches. It is associated with a credit cycle model introduced by Matsuyama, under the assumption of the Cobb-Douglas production function. We offer a detailed analysis of the dynamics of this family. In particular, using the skew tent map as a border collision normal form we obtain the conditions of abrupt transition from an attracting fixed point to an attracting cycle or a chaotic attractor (cyclic chaotic intervals). These conditions allow us to describe the bifurcation structure of the parameter space of the map in a neighborhood of the boundary related to the border collision bifurcation of the fixed point. Particular attention is devoted to codimension-two bifurcation points. Moreover, the described bifurcation structure confirms that the chaotic attractors of the considered map are robust, that is, persistent under parameter perturbations.     

Suppressing chaos of a complex Duffing's system using a random phase

The effect of random phase for a complex Duffing's system is investigated. We show as the intensity of random noise properly increases the chaotic dynamical behavior will be suppressed by the criterion of top Lyapunov exponent, which is computed based on the Khasminskii's formulation and the extension of Wedig's algorithm for linear stochastic systems. Also Poincaré map analysis, phase plot and the time evolution are carried out to confirm the obtained results of Lyapunov exponent on dynamical behavior including the stability, bifurcation and chaos. Thus excellent agreement between these results is found.     

Entropy of random chaotic interval map with noise which causes coarse-graining

A random chaotic interval map with noise which causes coarse-graining induces a finite-state Markov chain. For a map topologically conjugate to a piecewise-linear map with the Lebesgue measure being ergodic, we prove that the Shannon entropy for the induced Markov chain possesses a finite limit as the noise level tends to zero. In most cases, the limit turns out to be strictly greater than the Lyapunov exponent of the original map without noise.     

Existence and visualization of a hyperbolic structure in area-preserving maps

We present a simple method which displays a hyperbolic structure in the phase space of an area preserving map. Using the picture which this method yields one can estimate the entropy of the system and its Lyapunov exponent through the size of “hyperbolic cells” inside the chaotic region.     

动力系统实测数据的Lyapunov指数的矩阵算法

Lyapunov指数l是定量描述混沌吸引子的重要指标,自从1985年Wolf提出Lyapunov指数l的轨线算法以来,如何准确、快速地计算正的、最大的Lyapunov指数l_max便成为人们关注的问题,虽有不少成功计算的报导,但一般并不公开交流.在Zuo Bingwu理论算法的基础上,给出了Lyapunov指数l的具体的矩阵算法,并与Wolf的算法进行了比较,计算结果表明:算法能快速、准确地计算(主要是正的、最大的)Lyapunov指数l_max.并对Lyapunov指数l的大小所反应的吸引子的特性进行了分析,并得出了相应的结论.     

A comparative study of chaotic and white noise signals in digital watermarking

Digital watermarking is an ever increasing and important discipline, especially in the modern electronically-driven world. Watermarking aims to embed a piece of information into digital documents which their owner can use to prove that the document is theirs, at a later stage. In this paper, performance analysis of watermarking schemes is performed on white noise sequences and chaotic sequences for the purpose of watermark generation. Pseudorandom sequences are compared with chaotic sequences generated from the chaotic skew tent map. In particular, analysis is performed on highpass signals generated from both these watermark generation schemes, along with analysis on lowpass watermarks and white noise watermarks. This analysis focuses on the watermarked images after they have been subjected to common image distortion attacks. It is shown that signals generated from highpass chaotic signals have superior performance than highpass noise signals, in the presence of such attacks. It is also shown that watermarks generated from lowpass chaotic signals have superior performance over the other signal types analysed.     

Synchronization in pair-coupled maps with invariant measure

Chaos synchronization, as an important topic, has become an active research subject in non-linear science. By considering a symmetric two-dimensional map that possesses invariant measure in its diagonal and anti-diagonal invariant sub-manifolds, we have been able to introduce the most general pair-coupled map possessing invariant measure at synchronized or anti-synchronized states. Then chaotic synchronization and anti-synchronization are investigated in introduced model. We have calculated Kolmogrov–Sinai entropy and Lyapunov exponent as another tool to study the stability of pair-coupled map at synchronized and anti-synchronization states.     

On the error-sum function of tent map base series

We first introduce tent map base series. The tent map base series is special case of generalized Lüroth series which has the tent map as a base map. Then we study some elementary properties of its error-sum function, and show that the function is continuous.     

Generating a new chaotic attractor by feedback controlling method

A new chaotic system is found by feedback controlling method in this paper. According to the definition of the generalized Lorenz system, the new chaotic system does not belong to generalized Lorenz systems. We analyze the new system by means of phase portraits, Lyapunov exponents, fractional dimension, bifurcation diagram, and Poincaré map. The particular interest is that this novel system can generate two one‐scroll and one two‐scroll chaotic attractors with the variation of a single parameter. The obtained results show clearly that the system is a new chaotic system and deserves a further detailed investigation. Copyright © 2011 John Wiley & Sons, Ltd.     

A Continuous Archetype of Nonuniform Chaos in Area-Preserving Dynamical Systems

We propose a piecewise linear, area-preserving, continuous map of the two-torus as a prototype of nonlinear two-dimensional mixing transformations that preserve a smooth measure (e.g., the Lebesgue measure). The model lends itself to a closed-form analysis of both statistical and geometric properties. We show that the proposed model shares typical features that characterize chaotic dynamics associated with area-preserving nonlinear maps, namely, strict inequality between the line-stretching exponent and the Lyapunov exponent, the dispersive behavior of stretch-factor statistics, the singular spatial distribution of expanding and contracting fibers, and the sign-alternating property of cocycle dynamics. The closed-form knowledge of statistical and geometric properties (in particular of the invariant contracting and dilating directions) makes the proposed model a useful tool for investigating the relationship between stretching and folding in bounded chaotic systems, with potential applications in the fields of chaotic advection, fast dynamo, and quantum chaos theory.     

The explosive divergence in iterative maps of matrices

The effect of explosive divergence in generalized iterative maps of matrices is defined and described using formal algebraic techniques. It is shown that the effect of explosive divergence can be observed in an iterative map of square matrices of order 2 if and only if the matrix of initial conditions is a nilpotent matrix and the Lyapunov exponent of the corresponding scalar iterative map is greater than zero. Computational experiments with the logistic map and the circle map are used to illustrate the effect of explosive divergence occurring in iterative maps of matrices.     

Anti-control of continuous-time dynamical systems

Based on two basic characteristics of continuous-time autonomous chaotic systems, namely being globally bounded while having a positive Lyapunov exponent, this paper develops a universal and practical anti-control approach to design a general continuous-time autonomous chaotic system via Lyapunov exponent placement. This self-unified approach is verified by mathematical analysis and validated by several typical systems designs with simulations. Compared to the common trial-and-error methods, this approach is semi-analytical with feasible guidelines for design and implementation. Finally, using the Shilnikov criteria, it is proved that the new approach yields a heteroclinic orbit in a three-dimensional autonomous system, therefore the resulting system is indeed chaotic in the sense of Shilnikov.     

Dynamical phenomena occurring due to phase volume compression in nonholonomic model of the rattleback

We study numerically the dynamics of the rattleback, a rigid body with a convex surface on a rough horizontal plane, in dependence on the parameters, applying methods used earlier for treatment of dissipative dynamical systems, and adapted here for the nonholonomic model. Charts of dynamical regimes on the parameter plane of the total mechanical energy and the angle between the geometric and dynamic principal axes of the rigid body are presented. Characteristic structures in the parameter space, previously observed only for dissipative systems, are revealed. A method for calculating the full spectrum of Lyapunov exponents is developed and implemented. Analysis of the Lyapunov exponents of the nonholonomic model reveals two classes of chaotic regimes. For the model reduced to a 3D map, the first one corresponds to a strange attractor with one positive and two negative Lyapunov exponents, and the second to the chaotic dynamics of quasi-conservative type, when positive and negative Lyapunov exponents are close in magnitude, and the remaining exponent is close to zero. The transition to chaos through a sequence of period-doubling bifurcations relating to the Feigenbaum universality class is illustrated. Several examples of strange attractors are considered in detail. In particular, phase portraits as well as the Lyapunov exponents, the Fourier spectra, and fractal dimensions are presented.     

Study on the dynamical behaviors of a two-dimensional discrete system

The generalized Ushiki map is investigated. It is theoretically proven that there are transcritical and flip bifurcations and there exists a chaotic phenomenon in the sense of Marotto. And numerical simulations not only show the consistency with the theoretical analysis but also exhibit the complex dynamical behaviors including the period-6, 5, 3 windows, intermittency mechanics, attractor merging crisis and boundary crisis. The computation of Lyapunov exponents conforms the dynamical behaviors.