Novel image cryptosystem based on new 2D hyperchaotic map and dynamical chaotic S-box

Chaotic systems are widely used in image encryption due to their sensitivity to initial values, ergodicity, and other properties; many image encryption algorithms based on chaotic systems have been studied in the past few years. To obtain a more secure encryption algorithm, this work firstly proposes a new two-dimensional discrete hyperchaotic map, which has a wider continuous chaotic interval, larger Lyapunov exponents and passed all NIST and part of TestU01 tests. Then, we apply the proposed map to generate S-boxes and combine them in pairs; finally, twelve S-boxes are obtained, and the elements of the plaintext image are grouped, each group of pixels is summed, and modular operations are used to specify specific S-boxes. Next, each set of elements is bitwise XOR with the corresponding S-box. Finally, the cipher image is obtained by scrambling using chaotic signal. Experiments show that compared with some other encryption algorithms, the proposed S-box-based encryption method has higher security, and it resists to common attacks.

     

Designing a multi-scroll chaotic system by operating Logistic map with fractal process

Recently, chaotic systems have been widely investigated in several engineering applications. This paper presents a new chaotic system based on Julia’s fractal process, chaotic attractors and Logistic map in a complex set. Complex dynamic characteristics were analyzed, such as equilibrium points, bifurcation, Lyapunov exponents and chaotic behavior of the proposed chaotic system. As we know, one positive Lyapunov exponent proved the chaotic state. Numerical simulation shows a plethora of complex dynamic behaviors, which coexist with an antagonist form mixed of bifurcation and attractor. Then, we introduce an algorithm for image encryption based on chaotic system. The algorithm consists of two main stages: confusion and diffusion. Experimental results have proved that the proposed maps used are more complicated and they have a key space sufficiently large. The proposed image encryption algorithm is compared to other recent image encryption schemes by using different security analysis factors including differential attacks analysis, statistical tests, key space analysis, information entropy test and running time. The results demonstrated that the proposed image encryption scheme has better results in the level of security and speed.     

On stability analysis via Lyapunov exponents calculated from a time series using nonlinear mapping—a case study

The concept of Lyapunov exponents has been mainly used for analyzing chaotic systems, where at least one exponent is positive. The methods for calculating Lyapunov exponents based on a time series have been considered not reliable for computing negative and zero exponents, which prohibits their applications to potentially stable systems. It is believed that the local linear mapping leads to inaccurate matrices which prevent them from calculating negative exponents. In this work, the nonlinear approximation of the local neighborhood-to-neighborhood mapping is derived for constructing more accurate matrices. To illustrate the approach, the Lyapunov exponents for a stable balancing control system of a bipedal robot during standing are calculated. The time series is generated by computer simulations. Nonlinear mapping is constructed for calculating the whole spectrum of Lyapunov exponents. It is shown that, as compared with those from the linear mapping, (1) the accuracy of the negative exponents calculated using the nonlinear mapping is significantly improved; (2) their sensitivity to the time lag and the evolution time is significantly reduced; and (3) no spurious Lyapunov exponent is generated if the dimension of the state space is known. Thus, the work can contribute significantly to stability analysis of robotic control systems. Issues on extending the concept of Lyapunov exponents to analyzing stable systems are also addressed.     

Optimal image watermarking scheme based on chaotic map and quaternion wavelet transform

In this paper, an intelligent watermarking scheme optimized by adaptive differential evolution (ADE) is proposed based on chaotic map and quaternion wavelet transform (QWT). Watermark is first scrambled by a piecewise linear chaotic map to increase security and then inserted into the real QWT amplitude coefficients intelligently. ADE algorithm is explored to optimize the watermarking parameter (i.e., scaling factors) automatically. The experimental results demonstrate the high robustness against common signal processing operations, especially geometrical distortions of the proposed algorithm. Comparison results also indicate the superiority of the proposed algorithm over the existing watermarking algorithm.     

The analysis of the spectrum of Lyapunov exponents in a two-degree-of-freedom vibro-impact system

In the study of dynamical systems, the spectrum of Lyapunov exponents has been shown to be an efficient tool for analyzing periodic motions and chaos. So far, different calculating methods of Lyapunov exponents have been proposed. Recently, a new method using local mappings was given to compute the Lyapunov exponents in non-smooth dynamical systems. By the help of this method and the coordinates transformation proposed in this paper, we investigate a two-degree-of-freedom vibro-impact system with two components. For this concrete model, we construct the local mappings and the Poincaré mapping which are used to describe the algorithm for calculating the spectrum of Lyapunov exponents. The spectra of Lyapunov exponents for periodic motions and chaos are computed by the presented method. Moreover, the largest Lyapunov exponents are calculated in a large parameter range for the studied system. Numerical simulations show the success of the improved method in a kind of two-degree-of-freedom vibro-impact systems.     

结构可靠度FORM方法的混沌动力学分析

引入混沌动力学理论讨论了FORM收敛失败的非线性动力学根源. 给出了几个典 型函数在参数区间上的可靠指标分岔图，展示了极限状态函数经过FORM迭代成为 非线性映射后计算结果的周期振荡、分岔和混沌等复杂动力学现象，计算了非线性映射的 Lyapunov指数. 结果表明，极限状态函数设计点的曲率大小与FORM的收敛性没有简单的联 系，判别FORM迭代计算收敛性的指标是非线性映射的Lyapunov指数.     

A new chaotic system with fractional order and its projective synchronization

Based on Rikitake system, a new chaotic system is discussed. Some basic dynamical properties, such as equilibrium points, Lyapunov exponents, fractal dimension, Poincaré map, bifurcation diagrams and chaotic dynamical behaviors of the new chaotic system are studied, either numerically or analytically. The obtained results show clearly that the system discussed is a new chaotic system. By utilizing the fractional calculus theory and computer simulations, it is found that chaos exists in the new fractional-order three-dimensional system with order less than 3. The lowest order to yield chaos in this system is 2.733. The results are validated by the existence of one positive Lyapunov exponent and some phase diagrams. Further, based on the stability theory of the fractional-order system, projective synchronization of the new fractional-order chaotic system through designing the suitable nonlinear controller is investigated. The proposed method is rather simple and need not compute the conditional Lyapunov exponents. Numerical results are performed to verify the effectiveness of the presented synchronization scheme.     

A parallel image encryption algorithm based on the piecewise linear chaotic map and hyper-chaotic map

This paper proposes a parallel digital image encryption algorithm based on a piecewise linear chaotic map (PWLCM) and a four-dimensional hyper-chaotic map (FDHCM). Firstly, two decimals are obtained based on the plain-image and external keys, using a novel parallel quantification method. They are used as the initial value and control parameter for the PWLCM. Then, an encryption matrix and four chaotic sequences are constructed using the PWLCM and FDHCM, which control the permutation and diffusion processes. The proposed algorithm is implemented and tested in parallel based on a graphics processing unit device. Numerical analysis and experimental results show that the proposed algorithm achieves a high encryption speed and a good security performance, which provides a potential solution for real-time image encryption applications.     

Chaotic attitude motion of a magnetic rigid spacecraft

IntroductionAttitudedynamicsofspacecraftisascientificresearchsubjectwithgreatsignificance[1,2 ].Aschaosiswidelyanddeeplyinvestigated ,muchattentionhasbeenpaidonchaoticattitudemotionofspacecraft.Itnotonlyprovidesadefiniteengineeringbackgroundforexploringchaos,butalsooffersanewviewpointfordesigningspacecraft.Ithasbeenshownthatthereexistschaoticattitudemotioninsomemodelsofspacecraft,suchasspinningsatellitesinacircularorbit,gyrostatsatellitesinthegravitationalfield ,andtetheredsatellites[3 ,4].Ho…     

An efficient scheme for digital watermarking using chaotic map

This paper presents the new digital watermarking algorithm based on the chaotic map. Data can easily be hidden in images by using the least significant bits. In the anticipated algorithm, the logistic map is employed for locating embedding positions of chaotic watermark generation and a novel watermarking scheme is proposed. Simulation results reveal that the proposed technique is feasible and watermarks are indiscernible. The results illustrate that mean of energy, correlation, contrast, and homogeneity analyses of the proposed algorithm are quite optimal.     

Experimental Observation of Chaotic Motion in a Rotor with Rubbing

This paper presents an application for chaotic motion identification in a measured signal obtained in an experiment. The method of state space reconstruction with delay co-ordinates with the dynamic evolution described by a map is used. Poincaré diagrams, correlation dimensions and Lyapunov exponents are obtained as tools for deciding about the existence of chaotic behaviour. The method is applied to measurements of the lateral displacement of a vertical rotor experiencing rubbing and in some signals chaos is observed. The work concludes that the possibility of chaotic motion is well determined with the observation of Poincaré diagrams and computation of Lyapunov exponents. Correlation dimensions computations, strongly influenced by noise, are not convenient tools for investigation of chaotic behaviour in signals generated by mechanical systems.     

Chaotic attitude motion of a magnetic rigid spacecraft in an elliptic orbit and its control

This paper deals with the chaotic attitude motion of a magnetic rigid spacecraft with internal damping in an elliptic orbit. The dynamical model of the spacecraft is established. The Melnikov analysis is carried out to prove the existence of a complicated nonwandering Cantor set. The dynamical behaviors are numerically investigated by means of time history, Poincaré map, Lyapunov exponents and power spectrum. Numerical simulations demonstrate the chaotic motion of the system. The input-output feedback linearization method and its modified version are applied, respectively, to control the chaotic attitude motions to the given fixed point or periodic motion. The project supported by the National Natural Science Foundation of Chine (10082003)     

Modified Lyapunov exponent,new measure of dynamics

Lyapunov exponents, defined as exponential divergent or convergent rate of initially infinitely close solution trajectories, have been widely used for diagnosing chaotic systems, as well for stability analysis of nonlinear systems. Although calculated from the evolution of disturbance vectors associated with the flow, Lyapunov exponents are not associated with any specific directions, and such evolutions are driven by the dynamics in all directions in the state space. It is desirable to explore the asymptotic behaviors of the dynamic systems along certain specific directions and the specific dynamics driving such behaviors. In this paper, the Lyapunov exponents are modified. The modified Lyapunov exponents can indicate the exponential divergent or convergent rates in certain directions, which are driven by the dynamics in the same directions. The existence and the invariance to the initial conditions of the proposed modified exponents are proven mathematically. The algorithm for calculating the modified Lyapunov exponents from mathematical models is also developed. A wide range of case studies, from classical nonlinear dynamic systems to engineering systems, are presented to demonstrate the proposed modified Lyapunov exponents, and the indications of the modified exponents are also discussed. The proposed modified Lyapunov exponents can reveal additional insights into the system dynamics to the conventional Lyapunov exponents. Such information can be instrumental for stability control design.     

非线性函数的混沌优化方法比较研究

已有的混沌优化方法几乎都是利用Logistic映射作为混沌序列发生器，而Logistic映射产生的混沌序列的概率密度函数服从两头多、中间少的切比雪夫型分布，不利于搜索的效率和能力。为此，首先根据Logistie映射混沌轨道点密度函数的特点，建立改进的混沌-BFGS混合优化算法。之后，考虑到Kent映射混沌轨道点密度为均匀分布，建立了基于Kent映射的混沌-BFGS混合优化算法。然后对五种混合优化方法——不加改进的和改进的基于Logistic映射的混沌-BFGS法，基于Kent映射的混沌-BFGS法，Monte Carlo试验-BFGS法，网格-BFGS法进行了研究，分别对3个低维和2个高维非线性复杂测试函数进行优化计算，对它们的全局优化计算效率和寻优能力做了比较，并探讨了混合优化方法全局优化性能差异的原因。结果表明，混沌优化方法是与Monte Carlo方法类似的一种随机性试验优化方法。而且，这类优化方法的计算性能至少与以下因素有关：混沌／随机序列的统计性质，优化问题全局最优点位置。     

基于自适应格网数字水深模型的水下地形匹配定位算法

为了解决当前的水下地形匹配定位算法未考虑格网大小随海底地形变化而自动调整的问题,提出了一种基于自适应格网数字水深模型的水下地形匹配定位算法。首先,引入基于四叉树的自适应格网模型,改进其局部格网的相似性评估指标与构网约束方法;然后,设计匹配区的确定及待匹配航迹的选取策略,给出待匹配航迹的水深值的计算方法,构建出目标匹配定位的地形相关组合算子。实验结果表明:提出的算法相比于基于等距离的规则格网模型的算法有以下优势:1)在地形特征丰富区域的定位精度明显提高;2)可以避免出现地形特征越明显匹配定位精度越低的情况;3)能一定程度地克服水深系统误差对定位精度的影响。     

Competitive modes for the Baier–Sahle hyperchaotic flow in arbitrary dimensions

In this article, the stretch-twist-fold (STF) flow is numerically studied using phase portraits, sensitive dependence on initial conditions, Lyapunov exponents, power spectrum, and the Poincaré map. The stretch-twist-fold flow is a two-parameter family of Stokes flows defined in a unit sphere that is associated with the fluid particle motion that naturally arises in the dynamo theory, which proposes a mechanism by which celestial bodies, such as earth and sun can maintain and amplify the magnetic field continuously. For this continuous growth of magnetic field, scientists are interested to invent new tools for the nonfuel consumption magnetism propulsion for the low earth orbit of spacecrafts or satellites. General properties of a chaotic dynamical system reference to the stretch-twist-fold flow model are addressed and numerical solutions are generated to explain some of these properties. Analytically, we studied the local behavior at equilibrium points. The predictability of chaos in the STF flow with the numerical calculation of Lyapunov exponents and Poincaré map is presented in this paper.     

Complexity of chaotic binary sequence and precision of its numerical simulation

Numerical simulation is one of primary methods in which people study the property of chaotic systems. However, there is the effect of finite precision in all processors which can cause chaos to degenerate into a periodic function or a fixed point. If it is neglected the precision of a computer processor for the binary numerical calculations, the numerical simulation results may not be accurate due to the chaotic nature of the system under study. New and more accurate methods must be found. A quantitative computable method of sequence complexity evaluation is introduced in this paper. The effect of finite precision is evaluated from the viewpoint of sequence complexity. The simulation results show that the correlation function based on information entropy can effectively reflect the complexity of pseudorandom sequences generated by a chaotic system, and it is superior to the other measure methods based on entropy. The finite calculation precision of the processor has significant effect on the complexity of chaotic binary sequences generated by the Lorenz equation. The pseudorandom binary sequences with high complexity can be generated by a chaotic system as long as the suitable computational precision and quantification algorithm are selected and behave correctly. The new methodology helps to gain insight into systems that may exist in various application domains such as secure communications and spectrum management.     

Chaotic attitude motion of a magnetic rigid spacecraft and its control

This paper deals with chaotic attitude motion of a magnetic rigid spacecraft with internal damping in a circular orbit near the equatorial plane of the earth. The dynamical model of the problem is established. The Melnikov analysis is carried out to prove the existence of a complicated non-wandering Cantor set. The dynamical behaviors are numerically investigated by means of time history. Poincare map, power spectrum and Lyapunov exponents. Numerical simulations indicate that the onset of chaos is characterized by the intermittency as the increase of the torque of the magnetic forces and decrease of the damping. The input-output feedback linearization method is applied to control chaotic attitude motions to the given fixed point and periodic motion.     

A new pseudo-random number generator based on CML and chaotic iteration

In this paper, we propose a new algorithm of generating pseudorandom number generator (PRNG), which we call (couple map lattice based on discrete chaotic iteration (CMLDCI)) that combine the couple map lattice (CML) and chaotic iteration. And we can prove that this method can be written in a form of chaos map, which is under the sense of Devaney chaos. In addition, we test the new algorithm in NIST 800-22 statistical test suits and we use it in image encryption.     

Implementation of FPGA-based real time novel chaotic oscillator

Currently, chaotic systems and chaos-based applications are commonly used in the engineering fields. One of the main structures used in these applications is the chaos-based signal generators. Chaotic signal generators have an important role, particularly in chaotic communication and cryptology. In this study, the Pehlivan-Wei chaotic system, which is a recently developed chaotic system, has been implemented with FPGA using three distinct algorithms (the Euler, Heun, and RK4) for the first time in literature. Numerical and HDL approaches are implemented by these three algorithms to compare the performance of each model for use in chaotic generators. In addition, the Lyapunov exponents and phase portraits of the system have been extracted for chaos analysis. RMSE analysis has been conducted on the chaotic generators, which are modeled using the Euler, Heun, and RK4 algorithms in order to observe error rates of each numerical algorithm in a comparative aspect. The performance of new chaotic system with various data sets has been analyzed. The operation frequency of the chaotic oscillators synthesized and tested for the Virtex-6 FPGA chip has been able to reach up to 463.688 MHz and the chaotic system has been able to calculate 300,000 data sets in 0.0284 s. However, PC-based algorithm having highest performance score can calculate 300,000 data sets in a period of 75.363 s. A comparison study has been performed on the performance of the FPGA-based and PC-based solutions to evaluate each approach.