An Efficient Spectral Method-based Algorithm for Solving a ‎High-dimensional Chaotic Lorenz System

In recent years, many algorisms of realizing image encryption based on the chaotic method have been put forward to improve security during the process of image transmission. However in one-dimensional chaotic encryption system, the shortcomings of small secret key space and weak security are obvious. This paper aims to encrypt images in the space by employing high dimensional space chaotic system. Using “xor” and “mod” for computation, encrypt each pixel of the image one by one, then modify the parameter μ and ω in the space chaotic system through any simple functions g or h. Only one round of encryption is needed through this method, and the pixels of the image will be in chaos from all directions. As a result, the original image cannot be identified. The experiment results show that this encryption scheme has large secret key space and high security.

This - is a graduatelevel monographic textbook devoted to understanding, prediction and control of highdimensional chaotic and attractor systems of real life. The objective of the book is to provide - the serious reader with a serious scientific tool that will enable the actual performance of competitive research in highdimensional chaotic and attractor dynamics. The book has nine Chapters. The first Chapter gives a textbook-like introduction into the low-dimensional attractors and chaos. This Chapter has an inspirational character, similar to other books on nonlinear dynamics and deterministic chaos. The second Chapter deals with Smales topological transformations of stretching, squeezing and folding (of the systems phasespace), developed for the purpose of chaos theory. The third Chapter is devoted to Poincars 3-body problem and basic techniques of chaos control, mostly of Ott-Grebogi-Yorke type. The fourth Chapter is a review of both Landaus and topological phase transition theory, as well as Hakens synergetics. The fifth Chapter deals with phase synchronization in high-dimensional chaotic systems. The sixth Chapter presents high-tech Josephson junctions, the basic components for the future quantum computers. The seventh Chapter deals with fractals and fractional Hamiltonian dynamics. The 8th Chapter gives a review of modern techniques for dealing with turbulence, ranging from theparameterspace of the Lorenz attractor to the Lie symmetries. The last, 9th, Chapter attempts to give a brief on the cutting edge techniques of the high-dimensional nonlinear dynamics (including geometries, gauges and solitons, culminating into the chaos field theory).

The article aims to study the reduced-order anti-synchronization between projections of fractional order hyperchaotic and chaotic systems using active control method. The technique is successfully applied for the pair of systems viz., fractional order hyperchaotic Lorenz system and fractional order chaotic Genesio-Tesi system. The sufficient conditions for achieving anti-synchronization between these two systems are derived via the Laplace transformation theory. The fractional derivative is described in Caputo sense. Applying the fractional calculus theory and computer simulation technique, it is found that hyperchaos and chaos exists in the fractional order Lorenz system and fractional order Genesio-Tesi system with order less than 4 and 3 respectively. The lowest fractional orders of hyperchaotic Lorenz system and chaotic Genesio-Tesi system are 3.92 and 2.79 respectively. Numerical simulation results which are carried out using Adams-Bashforth-Moulton method, shows that the method is reliable and effective for reduced order anti-synchronization.

Discrete chaotic system, as a pseudo-random signal source, plays a very important role in realizing secure communication. However, many low-dimensional chaotic systems are prone to chaos degradation. Therefore, many scholars have studied the construction of high-dimensional chaotic systems. However, many existing algorithms for constructing high-dimensional chaotic systems have relatively high time complexity and relatively complex structures. To solve this problem, this paper explores an <i>n</i>-dimensional discrete hyperchaotic system with a simple structure. Firstly, the <i>n</i>-dimensional discrete hyperchaotic system is constructed by using sine function and power function and simple operations. Then, it is theoretically analyzed based on Jacobian matrix method that the system can have the positive Lyapunov exponents. Next, the algorithm time complexity, sample entropy, correlation dimension and other indexes are compared with those of the existing methods. The experimental results show that our system has a simple structure, high complexity and good algorithm time complexity. Therewith, a six-dimensional chaotic system is chosen as an example, and the phase diagram, bifurcation diagram, Lyapunov expnonents, complexity and other characteristics of the system are analyzed. The results show that the proposed system has good chaotic characteristics. Moreover, to show the application of the proposed system, we apply it to audio encryption. According to this system, we combine it with the XOR operation and true random numbers to explore a novel method of one-cipher audio encryption. Through experimental simulation, compared with some existing audio encryption algorithms, this algorithm can satisfy various statistical tests and resist various common attacks. It is also validated that the proposed system can be effectively applied to the field of audio encryption.

When chaotic systems are implemented on finite precision machines, it will lead to the problem of dynamical degradation. Aiming at this problem, most previous related works have been proposed to improve the dynamical degradation of low-dimensional chaotic maps. This paper presents a novel method to construct high-dimensional digital chaotic systems in the domain of finite computing precision. The model is proposed by coupling a high-dimensional digital system with a continuous chaotic system. A rigorous proof is given that the controlled digital system is chaotic in the sense of Devaney’s definition of chaos. Numerical experimental results for different high-dimensional digital systems indicate that the proposed method can overcome the degradation problem and construct high-dimensional digital chaos with complicated dynamical properties. Based on the construction method, a kind of pseudorandom number generator (PRNG) is also proposed as an application.

This paper proposes a user-biometric-based image encryption using chaotic dynamical systems, SHA-256 hash function, and zigzag transformation. Three chaotic systems, namely, the Henon map, logistic map, and Lorenz system, are used to apply the chaotic properties to the encryption system. The hash value of the user biometric image is used to generate the initial value of the Henon map. The initial values for the logistic map and Lorenz system are generated from the hash value of the plain image. SHA-256 hash algorithm generates the hash values of the biometric and plain images. The plain image is scrambled using the two-dimensional chaotic values generated by the Henon map first. Next, the first scrambled image’s pixels are XORed with the one-dimensional chaotic values generated from the logistic map and produce the second scrambled image. Finally, take the second scrambled image’s pixels, and XOR them with the three-dimensional chaotic values generated from the Lorenz system and create the final encrypted image. The robustness and flexibility of the encryption system are analyzed using various security and performance analyses. The experiment results are compared with the existing algorithms to prove the efficiency of the proposed algorithm.

The stability of a linear generalized synchronization approach between different chaot- ic systems with unknown parameters is investigated in this paper. An adaptive controller and some parameter update laws are proposed to achieve stable linear generalized synchronization and parame- ter identification of different chaotic systems. This novel synchronization method effectively unifies complete synchronization, anti-synchronization, projective synchronization, rotation synchronization and function synchronization. The main results are illustrated by two examples: the rotation synchro- nization between Lorenz and Chen system; the modified projective synchronization between Lorenz system and Lu system.

Controlling chaos in a high dimensional system with periodic parametric perturbations

This paper presents an Artificial Neural Network (ANN) design for a chaotic generator, and the training performances for a three layer ANN architecture with different number of hidden neurons. Chaotic systems can be synchronized and used for secure communication. Chaotic systems such as Lorenz attractor, Rossler attractor and Chen's system are generally implemented directly based on their definitions represented by a unique group of ordinary differential equations (ODEs). An feed forward ANN can be trained using the output values of a chaotic system. The training process is carried out on a computer and the weights are generated for all neurons in an ANN architecture. These weights are then used for a trained ANN architecture model to generate the expected output for the target chaotic system. The complexity of the ANN architecture defines the implementation cost and speed. Therefore it is beneficial to use less number of hidden neurons to achieve the target training performance. Lorenz attractor has its significance in studying chaotic systems and is used as the design subject in this paper. The 3-layer ANN has one input, one hidden and one output layer. The ANN architecture with 1 to 16 hidden neurons is designed and trained respectively using MATLAB Neural Network Toolbox with three training algorithms: Levenberg-Marquardt, Scaled Conjugate Gradient algorithm and Bayesian Regulation. The optimized ANN architecture can be used to improve the efficiency of the fixed-point implementation on an Field Programmable Gates Array (FPGA) device.

It is of vital significance to stabilize and control a chaotic system. But, since the control system always has excessive fluctuations which make them always perform badly in the process of stabilization, a control model that makes the unstable system stable without excessive fluctuations is hereby developed. In the control model, one piecewise continuous control is added in each of the two halves of every control period. Then two impulses: one at the middle and another at the end of the control period of the system. Based on the Lyapunov's method, when the newly developed model is introduced into an appropriate chaotic system, it is proved to make the system stable. Precisely, the model can be applied to chaotic Chua's system and Lorenz system as examples to prove the validity of the results. Compared with other current control models, this model proves to control the unstable system more effectively in that it is more general and saves time.

Multi-switching combination synchronization of three different chaotic systems via nonlinear control

This paper investigates the global synchronization method of two identical systems and two different chaotic systems. The proposed method is effective and convenient to synchronize two identical systems and two different chaotic systems by using nonlinear control functions. The method has been applied successfully to make two identical four-scroll attractor and also two different chaotic systems; four-scroll attractor and Lorenz system globally synchronized. Numerical simulations are given to validate the proposed synchronization methods. Key words: Chaotic system, global synchronization, four-scroll attractor,Lorenz system, Lyapunov function.

Recently, many continuous-time chaotic systems were synthesized using microcontrollers and FPGAs. This requires applying mathematical discretization to convert integration into recursion. Depending on the approximation algorithms, the speed of the numerical processors, and the number of bits used to represent data, different accuracies and stabilities could be obtained. This article explores the conditions necessary to faithfully generate signals that reflect the true behavior of the chaotic systems, while maintaining the same values for their Lya-punov exponents. This is very important for chaos control and synchronization, especially for applications in secure communication that rely on digital cryptography. The Lorenz system and the Duffing oscillator are investigated to illustrate the effect of having autonomous versus non-autonomous structures. In addition, the Nosé-Hoover dynamical model is investigated to detect the relationship between agility and numerical accuracy. The obtained results prove that identical deterministic chaotic systems can behave differently, for the same set of initial conditions, depending on the discretization algorithm. This added sensitivity necessitates careful design of the mathematical models required for digital implementations of chaotic systems. The article concludes with useful recommendations for best practices in designing, synthesizing, and implementing digital chaotic systems, while commenting on the best compromise between mathematical complexity and numerical accuracy.

Considering that chaotic systems are highly sensitive to parameters, we design two new parameter variable chaotic systems by constructing parameter perturbation items. These systems are constructed using the state variables of the Liu chaotic system to perturb the parameters of the Lorenz and Chen chaotic systems and are called the Lorenz-Liu chaotic system (LLCS) and Chen-Liu chaotic system (CLCS), respectively. In particular, the parameter perturbation items constructed in this study are not periodic but rather chaotic signals and change in real time. Compared with the original systems, they exhibit more complex randomness and dynamic behaviors. In the proposed cryptosystem, which considers the concept of Deoxyribonucleic Acid (DNA), the solar radio spectrogram is dynamically encoded through the LLCS, and then, the CLCS is used to scramble and diffuse the decoding matrices. In addition, the algorithm uses the 256-bit Secure Hash Algorithm (SHA-256) to generate the initial keys, which enhances the algorithm’s sensitivity to plaintext. Simulation results and security analysis show that the cryptosystem has a large key space and high key sensitivity, and can resist various attacks, such as differential attacks and chosen-plaintext attacks.

This study deals with the relevant and important area of many fields of mathematics and physics - chaotic systems. Three modified systems of Chua differential equations were considered, and the chaotic structure of their solutions was compared with the structure of solutions of classical Lorentz and Rössler chaotic systems. The following methods were used to achieve the set goal: the Runge-Kutta method, building a phase portrait, determining Lyapunov exponents and noise level, and comparative analysis. A detailed analysis of the structure of chaotic solutions of various differential equations was carried out. It was established that the chaotic solution’s structure depends on the differential equation’s properties and the initial conditions. According to the obtained results, one of the modifications of the Chua system is significantly superior to classical chaotic systems and can be used as a chaos generator. Prospects for further research involve expanding the scope of the study and the generalization of the obtained results for a wider class of systems of differential equations.