Universal encoding for unimodal maps

The standard encoding procedure to describe the chaotic orbits of unimodal maps is accurately investigated. We show that the grammatical rules of the underlying language can be easily classified in a compact form by means of a universal parameter . Two procedures to construct finite graphs which approximate non-Markovian cases are discussed, showing also the intimate relation with the corresponding construction of approximate Markov partitions. The convergence of the partial estimates of the topological entropy is discussed, proving that the error decreases exponentially with the length of the sequences considered. The rate is shown to coincide with the topological entropyh itself. Finally, a superconvergent method to estimateh is introduced.     

Bifurcation frequency for unimodal maps

We consider some natural one-parameter unfoldingsf , of a unimodal mapf ₀ whose periodic points are hyperbolic and whose critical point is nondegenerate and eventually periodic. Among other facts, it follows from our theorems that, if the Julia set off ₀ does not contain intervals, the relative measure of the bifurcation set is zero at zero.     

Invariants and chaotic maps

A one-dimensional mapf(x) is called an invariant of a two-dimensional mapg(x, y) ifg(x, f(x))=f(f(x)). The logistic map is an invariant of a class of two-dimensional maps. We construct a class of two-dimensional maps which admit the logistic maps as their invariant. Moreover, we calculate their Lyapunov exponents. We show that the two-dimensional map can show hyperchaotic behavior.     

Noise-resistant chaotic maps

Synchronized chaotic systems are highly vulnerable to noise added to the synchronizing signal. It was previously shown that chaotic circuits could be built that were less sensitive to this type of noise. In this work, simple chaotic maps are demonstrated that are also less sensitive to added noise. These maps are based on coupling a shift map to a digital filter. These maps are simple enough that they should help lead to an understanding of how noise-robust chaotic systems work. (c) 2002 American Institute of Physics.     

A secure key agreement protocol based on chaotic maps

To guarantee the security of communication in the public channel,many key agreement protocols have been proposed.Recently,Gong et al.proposed a key agreement protocol based on chaotic maps with password sharing.In this paper,Gong et al.’s protocol is analyzed,and we find that this protocol exhibits key management issues and potential security problems.Furthermore,the paper presents a new key agreement protocol based on enhanced Chebyshev polynomials to overcome these problems.Through our analysis,our key agreement protocol not only provides mutual authentication and the ability to resist a variety of common attacks,but also solve the problems of key management and security issues existing in Gong et al.’s protocol.     

Binary tree approach to scaling in unimodal maps

Ge, Rusjan, and Zweifel introduced a binary tree which represents all the periodic windows in the chaotic regime of iterated one-dimensional unimodal maps. We consider the scaling behavior in a modified tree which takes into account the self-similarity of the window structure. A nonuniversal geometric convergence of the associated superstable parameter values towards a Misiurewicz point is observed for almost all binary sequences with periodic tails. For these sequences the window period grows arithmetically down the binary tree. There are an infinite number of exceptional sequences, however, for which the growth of the window period is faster. Numerical studies with a quadratic maximum suggest more rapid than geometric scaling of the superstable parameter values for such sequences.     

Labyrinthine pathways towards supercycle attractors in unimodal maps

As an important preceding step for the demonstration of an uncharacteristic (q-deformed) statisticalmechanical structure in the dynamics of the Feigenbaum attractor we uncover previously unknown properties of the family of periodic superstable cycles in unimodal maps. Amongst the main novel properties are the following: i) The basins of attraction for the phases of the cycles develop fractal boundaries of increasing complexity as the period-doubling structure advances towards the transition to chaos. ii) The fractal boundaries, formed by the pre-images of the repellor, display hierarchical structures organized according to exponential clusterings that manifest in the dynamics as sensitivity to the final state and transient chaos. iii) There is a functional composition renormalization group (RG) fixed-point map associated with the family of supercycles. iv) This map is given in closed form by the same kind of q-exponential function found for both the pitchfork and tangent bifurcation attractors. v) There is final-stage ultra-fast dynamics towards the attractor, with a sensitivity to initial conditions which decreases as an exponential of an exponential of time. We discuss the relevance of these properties to the comprehension of the discrete scale-invariance features, and to the identification of the statistical-mechanical framework present at the period-doubling transition to chaos. This is the first of three studies (the other two are quoted in the text) which together lead to a definite conclusion about the applicability of q-statistics to the dynamics associated to the Feigenbaum attractor.      

A novel bit-level image encryption algorithm based on chaotic maps

Recently, a number of chaos-based image encryption algorithms have been proposed at the pixel level, but little research at the bit level has been conducted. This paper presents a novel bit-level image encryption algorithm that is based on piecewise linear chaotic maps (PWLCM). First, the plain image is transformed into two binary sequences of the same size. Second, a new diffusion strategy is introduced to diffuse the two sequences mutually. Then, we swap the binary elements in the two sequences by the control of a chaotic map, which can permute the bits in one bitplane into any other bitplane. The proposed algorithm has excellent encryption performance with only one round. The simulation results and performance analysis show that the proposed algorithm is both secure and reliable for image encryption.     

Renormalization of binary trees derived from one-dimensional unimodal maps

For one-dimensional unimodal mapsh (x):I I, whereI=[x ₀,x ₁] when =_max, a binary tree which includes all the periodic windows in the chaotic regime is constructed. By associating each element in the tree with the superstable parameter value of the corresponding periodic interval, we define a different unimodal map. After applying a certain renormalization procedure to this new unimodal map, we find the period-doubling fixed point and the scaling constant. The period-doubling fixed point depends on the details of the maph (x), whereas the scaling constant equals the derivative . The thermodynamics and the scaling function of the resulting dynamical system are also discussed. In addition, the total measure of the periodic windows is calculated with results in basic agreement with those obtained previously by Farmer. Up to 13 levels of the tree have been included, and the convergence of the partial sums of the measure is shown explicitly. A new scaling law has been observed, i.e., the product of the length of a periodic interval characterized by sequenceQ and the scaling constant ofQ is found to be approximately 1.     

一种混沌映射的相空间去噪方法

对于混沌映射来说,它们的频谱比混沌流的频谱更广阔,与噪声频谱的重叠率更高,所以混沌流的去噪方法对它们并不适用. 在半盲分析法的框架下,混沌系统的参数估计问题终将归结为最小二乘估计问题. 本文从最小二乘拟合的角度出发估计混沌映射的演化参数,进而通过相空间重构以及投影操作,实现对观测信号的噪声抑制. 实验结果表明,该算法的去噪效果优于扩展卡尔曼滤波器（extended Kalman filter,EKF）和无先导卡尔曼滤波器（unscneted Kalman filter,UKF）,并且能够较大程度地将信号源的混沌特征量还原. 关键词： 混沌 噪声抑制 相空间重构 投影     

Generalized complexity measures and chaotic maps

The logistic and Tinkerbell maps are studied with the recently introduced generalized complexity measure. The generalized complexity detects periodic windows. Moreover, it recognizes the intersection of periodic branches of the bifurcation diagram. It also reflects the fractal character of the chaotic dynamics. There are cases where the complexity plot shows changes that cannot be seen in the bifurcation diagram.     

Regular approximations to chaotic maps

We construct regular analytic approximations to partly chaotic maps on a two-dimensional torus-the Standard Map in particular. Possible extensions are discussed.     

Design new chaotic maps based on dimension expansion

Based on the high-dimensional(HD) chaotic maps and the sine function, a new methodology of designing new chaotic maps using dimension expansion is proposed. This method accepts N dimensions of any existing HD chaotic map as inputs to generate new dimensions based on the combined results of those inputs. The main principle of the proposed method is to combine the results of the input dimensions, and then performs a sine-transformation on them to generate new dimensions.The characteristics of the generated dimensions are totally different compared to the input dimensions. Thus, both of the generated dimensions and the input dimensions are used to create a new HD chaotic map. An example is illustrated using one of the existing HD chaotic maps. Results show that the generated dimensions have better chaotic performance and higher complexity compared to the input dimensions. Results also show that, in the most cases, the generated dimensions can obtain robust chaos which makes them attractive to usage in a different practical application.     

Image encryption based on new Beta chaotic maps

In this paper, we created new chaotic maps based on Beta function. The use of these maps is to generate chaotic sequences. Those sequences were used in the encryption scheme. The proposed process is divided into three stages: Permutation, Diffusion and Substitution. The generation of different pseudo random sequences was carried out to shuffle the position of the image pixels and to confuse the relationship between the encrypted the original image, so that significantly increasing the resistance to attacks. The acquired results of the different types of analysis indicate that the proposed method has high sensitivity and security compared to previous schemes.     

Generalized entropies of chaotic maps and flows: A unified approach

A thermodynamic study of nonlinear dynamical systems, based on the orbits' return times to the elements of a generating partition, is proposed. Its grand canonical nature makes it suitable for application to both maps and flows, including autonomous ones. When specialized to the evaluation of the generalized entropies K(q), this technique reproduces a well-known formula for the metric entropy K(1) and clarifies the relationship between a flow and the associated Poincare maps, beyond the straightforward case of periodically forced nonautonomous systems. Numerical estimates of the topological and metric entropy are presented for the Lorenz and Rossler systems. The analysis has been carried out exclusively by embedding scalar time series, ignoring any further knowledge about the systems, in order to illustrate its usefulness for experimental signals as well. Approximations to the generating partitions have been constructed by locating the unstable periodic orbits of the systems up to order 9. The results agree with independent estimates obtained from suitable averages of the local expansion rates along the unstable manifolds. (c) 1997 American Institute of Physics.     

混沌的乘法规律

从理论上分析了几个混沌映射相乘的规律，提出了混沌相乘的思想.数值仿真与理论计算结果表明，几个混沌映射相乘仍然具有非线性动力学特性和分岔序列.并且具有混沌吸引子和对初值敏感性.混沌的乘法概念具有重要的理论意义和应用价值. 关键词： 相乘混沌 吸引子 分岔序列     

Synchronization learning of coupled chaotic maps

We study the dynamics of an ensemble of globally coupled chaotic logistic maps under the action of a learning algorithm aimed at driving the system from incoherent collective evolution to a state of spontaneous full synchronization. Numerical calculations reveal a sharp transition between regimes of unsuccessful and successful learning as the algorithm stiffness grows. In the regime of successful learning, an optimal value of the stiffness is found for which the learning time is minimal.