Dimensional Hyperchaotic System Research Articles

Dynamics and Adaptive Control of a Novel 5D Hyperchaotic System: Either Hidden Attractor or Self-excited with Unusual Nature of Unstable Equilibria

In 2020, J. Sprott proposed a new three dimensional chaotic system with special features such like 1) dissipative and time-reversible; 2) no equilibrium point; 3) lien of initial conditions goes to the attractor. We observed that an extension of the so-called Sprott's 2020 system to four dimensional system with complex dynamics showed either chaotic or hyperchaotic with unbounded orbits. In this paper, a novel five dimensional hyperchaotic system based on Sprott's 2020 system has been proposed. The proposed system shows complex dynamics like hyperchaotic. The proposed system can be classified as a hidden attractor where no equilibrium point appeared or self-excited where an unusual nature of unstable equilibrium points connected to a very complicated function called Lambert W appeared. The dynamical properties of such system are discovered by computing the Lyapunov exponents and bifurcation diagram. Adaptive control to the proposed system was provided.

Design of two dimensional hyperchaotic system through optimization benchmark function

The hyperchaotic systems are essentially needed for various applications, especially multimedia encryption, watermarking and communications. However, existing chaotic systems have limited chaotic performance in terms of precise chaos measuring tools like bifurcation and attractor diagrams, Lyapunov exponent (LE), 0-1 test, correlation dimension (CD) and Kolmogorov entropy (KE). In this paper, a new hyperchaotic system so-called 2D Rosenbrock map is designed by exploiting the Rosenbrock function, which has perfect swinging characteristics in modular form. In order to manage the map, two control parameters are inserted to the Rosenbrock function. The proposed 2D Rosenbrock map is self-verified and also validated over a comparison with the recently reported results. The 2D Rosenbrock map has excellent ergodicity and diversity properties. Moreover, the 2D Rosenbrock map is implemented to multimedia encryption. The findings manifest that the designed 2D Rosenbrock map owns superior chaotic capability due to its reproduction and oscillation features.

Internal Synchronization Using Adaptive Control

This paper mainly deals on the issue of a chaotic synchronization of a master and slave systems. It is generally the requirement of the synchronization that someone needs at least one to one master and slave systems. In the current study, the authors introduce the concept of a synchronization in which there is no need of slave/response system externally. Furthermore, the synchronization has been demonstrated here within a system among the subsystems of different orders. In addition, adaptive control is chosen for the synchronization among various combinations in multiswitching manner. For demonstration purpose, Lorenz Six Dimensional Hyper Chaotic System (L6DHCS) is chosen. There are three different kinds of possible switches presented by the authors formed within the considered system. The numerical simulations are carried out to validate the effectiveness of the analytical technique using Mathematica.

Four dimensional hyperchaotic communication system based on dynamic feedback synchronization technique for image encryption systems

<span>This paper presents the design and simulation of a hyperchaotic communication system based on four dimensions (4D) Lorenz generator. The synchronization technique that used between the master/transmitter and the slave/receiver is based on dynamic feedback modulation technique (DFM). The mismatch error between the master dynamics and slave dynamics are calculated continuously to maintain the sync process. The information signal (binary image) is masked (encrypted) by the hyperchaotic sample x of Lorenz generator. The design and simulation of the overall system are carried out using MATLAB Simulink software. The simulation results prove that the system is suitable for securing the plain-data, in particular the image data with a size of 128×128 pixels within 0.1 second required for encryption, and decryption in the presence of the channel noise. The decryption results for gray and colored images show that the system can accurately decipher the ciphered image, but with low level distortion in the image pixels due to the channel noise. These results make the proposed cryptosystem suitable for real time secure communications.</span>

A novel 4 dimensional hyperchaotic system with its control, Synchronization and Implementation

This paper presents a new hyperchaotic system which shows some interesting features, the system is 4-dimensional with 4 nonlinearities. An extensive numerical analysis has showed that the system has some interesting features and strange behaviors. The numerical analysis includes studying the effect of system parameters and initial conditions. Some of the important properties of the system with parameter set, in which the system is hyperchaotic, such as Lyapunov exponents and Lyapunov dimension, dissipation and symmetry are found and discussed. In the next part of our work, a tracking controller for the proposed system is designed and then a synchronization control system for two identical systems is designed. The design procedure uses combination of a simple synergetic control with adaptive updating laws to identify the unknown parameters derived basing on Lyapunov theorem. Hardware implementation based on microcontroller unit (MCU) board is proposed and tested and used to experimentally validate the designed control and synchronization systems. As an application, the designed synchronization system is used as a secure analogue communication system. Using MATLAB, Simulation study for the control and synchronization systems is presented. The simulation and experimental study have been showed excellent results.

Design and Realization of a Hyperchaotic Memristive System for Communication System on FPGA

In this study, a memristor based hyperchaotic circuit is presented and implemented for communication systems on FPGA platform. Four dimensional hyperchaotic system, which contains active flux controlled memristor is designed by using a smooth continuous nonlinearity. Dynamical characteristics of designed hyperchaotic circuit are examined such as equilibrium points, chaotic attractors, Lyapunov exponents and bifurcation diagram. Furthermore, an electronic circuit model of hyperchaotic system has been modeled and results are submitted. Chaotic circuits are used in communication systems especially in secure communication due to their sensitive dependence on the initial conditions, not periodic, and having a spread spectrum. By using nonlinearity of memristor, the signals obtained from memristor based hyperchaotic system have been realized to analog and digital communication schemes on FPGA platform, which is suitable for re-programmable and reconfigurable systems. The success of memristor based hyperchaotic circuit with FPGA based communication is demonstrated by both simulation and experimental results.

Integrated Partial Encryption and Watermarking Technique for Digital Video using Chaos and Support Images

Encryption and Watermarking are image processing tools that are used mostly separately. Encrypting a document completely obscures it and watermarking is applied mostly for copy right protection. For entertainment type videos, partial encryption and watermarking may be integrated such that both objectives are attained. We report in this paper a novel combined partial encryption and watermarking algorithm for digital videos using chaotic system and support image. Watermarks are generated by using iris information of the authentic user or the authentic producer of the digital video. Watermarks of full size and half size are embedded in the luminance frame (s) of the video using simple LSB steganography. A sequence of color frames is watermarked in two phases, and partially encrypted in two layers, the watermarks considered are encrypted before embedding. The luminance frame (s) is (are) preprocessed before embedding with the watermarks to defy stega-analysis. Chaotic logistic map and a gray level support digital image is used for obtaining unique row/column sequences as well as for generating robust random binary sequences. A support color digital image is used in the first layer encryption of the digital video. Modified Rabinovich 4 dimensional hyper-chaotic system is used for second layer encryption. Encryption algorithm is applied in R, G, B frames separately. The technique may also be applied for integrated encryption and watermarking of digital images.

Generalized Combination Synchronization of Three Different Dimensional Fractional Chaotic and Hyperchaotic Systems Using Three Scaling Matrices

Generalized Combination Synchronization of Three Different Dimensional Fractional Chaotic and Hyperchaotic Systems Using Three Scaling Matrices

Construction of Higher-Dimensional Hyperchaotic Systems with a Maximum Number of Positive Lyapunov Exponents under Average Eigenvalue Criteria

Over the last 40 years, the design of [Formula: see text]-dimensional hyperchaotic systems with a maximum number ([Formula: see text]) of positive Lyapunov exponents has been an open problem for research. Nowadays it is not difficult to design [Formula: see text]-dimensional hyperchaotic systems with less than ([Formula: see text]) positive Lyapunov exponents, but it is still extremely difficult to design an [Formula: see text]-dimensional hyperchaotic system with the maximum number ([Formula: see text]) of positive Lyapunov exponents. This paper aims to resolve this challenging problem by developing a chaotification approach using average eigenvalue criteria. The approach consists of four steps: (i) a globally bounded controlled system is designed based on an asymptotically stable nominal system with a uniformly bounded controller; (ii) a closed-loop pole assignment technique is utilized to ensure that the numbers of eigenvalues with positive real parts of the controlled system be equal to ([Formula: see text]) and ([Formula: see text]), respectively, at two saddle-focus equilibrium points; (iii) the number of average eigenvalues with positive real parts is ensured to be equal to ([Formula: see text]) for the controlled system over a given control period; (iv) the smallest value of the positive real parts of the average eigenvalues is ensured to be greater than a given threshold value. Finally, the paper is closed with some typical examples which illustrate the feasibility and performance of the proposed design methodology.

A Novel Image Encryption Scheme Based on 7D Hyperchaotic System and Row-column Simultaneous Swapping

In this paper, a novel image encryption algorithm is proposed based on a seven- dimensional (7D) hyperchaotic system and simultaneous row-column swapping. First, the 7D hyperchaotic system is introduced. The SHA-512 hash function is applied in order to generate the system parameters and initial values of the 7D hyperchaotic system. Seven real numbers are transformed in order to form three new sequences. The new sequences are used for the scrambling and diffusion operations. Then, the scrambling matrix is formed, and a plain image is subjected to the permutation process. Finally, the diffusion method is performed on the scrambled image, and the cipher image is ultimately obtained. The experimental results reveal that the proposed scheme has good performance. This scheme has large secret keys and is highly sensitive to the plain images and initial keys. This method could also resist many kinds of attacks. The proposed method is superior with respect to its security and encryption performance compared with some other methods.

Image compression-encryption algorithms by combining hyper-chaotic system with discrete fractional random transform

Based on hyper-chaotic system and discrete fractional random transform, an image compression-encryption algorithm is designed. The original image is first transformed into a spectrum by the discrete cosine transform and the resulting spectrum is compressed according to the method of spectrum cutting. The random matrix of the discrete fractional random transform is controlled by a chaotic sequence originated from the high dimensional hyper-chaotic system. Then the compressed spectrum is encrypted by the discrete fractional random transform. The order of DFrRT and the parameters of the hyper-chaotic system are the main keys of this image compression and encryption algorithm. The proposed algorithm can compress and encrypt image signal, especially can encrypt multiple images once. To achieve the compression of multiple images, the images are transformed into spectra by the discrete cosine transform, and then the spectra are incised and spliced into a composite spectrum by Zigzag scanning. Simulation results demonstrate that the proposed image compression and encryption algorithm is of high security and good compression performance.

Parameter estimation of fractional-order arbitrary dimensional hyperchaotic systems via a hybrid adaptive artificial bee colony algorithm with simulated annealing algorithm

Hyperchaos can be observed in fractional-order nonlinear systems with suitable orders. The knowledge about systematic parameters and fractional orders is important for control and synchronization of fractional-order hyperchaotic systems. In this article, parameter estimation of fractional-order arbitrary dimensional hyperchaotic systems is investigated. Firstly, estimation of systematic parameters and fractional orders is formulated as a multi-dimensional optimization problem by treating the fractional orders as additional parameters. Secondly, a novel method called hybrid adaptive artificial bee colony algorithm with simulated annealing algorithm is proposed to deal with this optimization problem. Finally, numerical simulations and comparisons with other typical algorithms are done to demonstrate the effectiveness of the proposed algorithm, which provides a promising tool for estimation of fractional-order arbitrary dimensional hyperchaotic systems as well as other numerical optimization problems in different fields.

Computing two dimensional Poincaré maps for hyperchaotic dynamics

Poincaré map (PM) is one of the felicitous discrete approximation of the continuous dynamics. To compute PM, the discrete relation(s) between the successive point of interactions of the trajectories on the suitable Poincaré section (PS) are found out. These discrete relations act as an amanuensis of the nature of the continuous dynamics. In this article, we propose a computational scheme to find a hyperchaotic PM (HPM) from an equivalent three dimensional (3D) subsystem of a 4D (or higher) hyperchaotic model. For the experimental purpose, a standard four dimensional (4D) hyperchaotic Lorenz-Stenflo system (HLSS) and a five dimensional (5D) hyperchaotic laser model (HLM) is considered. Equivalent 3D subsystem is obtained by comparing the movements of the trajectories of the original hyperchaotic systems with all of their 3D subsystems. The quantitative measurement of this comparison is made promising by recurrence quantification analysis (RQA). Various two dimensional (2D) Poincaré mas are computed for several suitable Poincaré sections for both the systems. But, only some of them are hyperchaotic in nature. The hyperchaotic behavior is verified by positive values of both one dimensional (1D) Lyapunov Exponent (LE-I) and 2D Lyapunov Exponent (LE-II). At the end, similarity of the dynamics between the hyperchaotic systems and their 2D hyperchaotic Poincaré maps (HPM) has been established through mean recurrence time (MRT) statistics for both of 4D HLSS and 5D HLM and the best approximated discrete dynamics for both the hyperchaotic systems are found out.

Single input sliding mode control for hyperchaotic Lu system with parameter uncertainty

In this paper, design of sliding mode controller (SMC) is presented to investigate the stabilization, complete synchronization and adaptive synchronization of four dimensional hyperchaotic Lu systems with parameter uncertainty. To achieve this goal, sliding mode control scheme along with Lyapunov stability theory is utilized. A proportional integral switching surface is proposed to ensure the stability of the closed-loop system in sliding motion. The SMC has been proposed to guarantee the occurrence of the sliding motion. It has also been shown that by proper choice of the adaptation laws for parameters, systems can be synchronized in conventional manner in master-slave configuration, in uncertain environment. The proposed adaptation laws also ensure the convergence of uncertain parameters to their true value in all the cases. Finally, numerical simulations are performed to demonstrate the effectiveness of the proposed approach.

Design and Application of a NEW Seven-Dimensional Hyperchaotic System

Because the chaotic characteristics of the hyperchaotic system is more complex, so design a higher dimensional hyperchaotic system has become a new orientation of the chaos theory research. This article construct a seven-dimensional third-order hyperchaotic system, this syetem is proved to be hyperchaotic through the MATLAB simulation and the Lyapunov exponential calculation. This article also design the realizing hardware circuit and do its simulation with Multisim. The MATLAB simulation results are consistent with the Multisim simulation results for the designed system and have the same chaotic attractor, which shows the realizability of the system. The chaotic signal source is used for the image encryption and decryption in order to ensure the confidential transmission of images.

Hyperchaos synchronization of fractional-order arbitrary dimensional dynamical systems via modified sliding mode control

This paper considers the design of adaptive sliding mode control approach for synchronization of a class of fractional-order arbitrary dimensional hyperchaotic systems with unknown bounded disturbances. This approach is based on the principle of sliding mode control and adaptive compensation term for solving the problem of synchronization of the unknown parameters in fractional-order nonlinear systems. In particular, a novel fractional-order five dimensional hyperchaotic system has been introduced as a representative example. Furthermore, global stability and asymptotic synchronization between the outputs of master and slave systems can be achieved based on the modified Lyapunov functional and fractional stability condition. Simulation results are provided in detail to illustrate the performance of the proposed approach.

A Novel Four-Wing Hyperchaotic Complex System and Its Complex Modified Hybrid Projective Synchronization with Different Dimensions

We introduce a new Dadras system with complex variables which can exhibit both four-wing hyperchaotic and chaotic attractors. Some dynamic properties of the system have been described including Lyapunov exponents, fractal dimensions, and Poincaré maps. More importantly, we focus on a new type of synchronization method of modified hybrid project synchronization with complex transformation matrix (CMHPS) for different dimensional hyperchaotic and chaotic complex systems with complex parameters, where the drive and response systems can be asymptotically synchronized up to a desired complex transformation matrix, not a diagonal matrix. Furthermore, CMHPS between the novel hyperchaotic Dadras complex system and other two different dimensional complex chaotic systems is provided as an example to discuss increased order synchronization and reduced order synchronization, respectively. Numerical results verify the feasibility and effectiveness of the presented schemes.

A New Five Dimensional Hyperchaotic System and its Fractional Order Form

In this paper, a new five-dimensional hyperchaotic system by introducing two additional states feedback into a three-dimensional smooth chaotic system. With three nonlinearities, this system has more than one positive Lyapunov exponents. Based on the fractional derivative theory, the fractional-order form of this new hyperchaotic system has been investigated. Through predictor-corrector algorithm, the system is proved by numerical simulation analysis. Simulation results are provided to illustrate the performance of the fractional-order hyperchaotic attractors well.

Pitchfork bifurcation and circuit implementation of a novel Chen hyper-chaotic system

In this paper, a novel four dimensional hyper-chaotic system is coined based on the Chen system, which contains two quadratic terms and five system parameters. The proposed system can generate a hyper-chaotic attractor in wide parameters regions. By using the center manifold theorem and the local bifurcation theory, a pitchfork bifurcation is demonstrated to arise at the zero equilibrium point. Numerical analysis demonstrates that the hyper-chaotic system can generate complex dynamical behaviors, e.g., a direct transition from quasi-periodic behavior to hyper-chaotic behavior. Finally, an electronic circuit is designed to implement the hyper-chaotic system, the experimental results are consist with the numerical simulations, which verifies the existence of the hyper-chaotic attractor. Due to the complex dynamic behaviors, this new hyper-chaotic system is useful in the secure communication.

Two-Dimensional Hyper-Chaotic Mapping Interval Newton Iterative Method to Mechanism Synthesis

Mechanism synthesis questions can be transformed into nonlinear equations to be found. Interval Newton iterative method is an important technique to one dimensional and multidimensional variables and iterative process exhibits sensitive dependence on initial guess point. The characteristic of hyper-chaotic sequences produced by two dimensional hyper-chaotic discrete systems was analyzed. Making use of the advantage of giving rigorous bounds for the exact solution, for the first time, combining hyper-chaos sequences and interval Newton iteration with Krawczyk operator, a new method to find all solutions was proposed. The numerical examples in linkage synthesis and approximate synthesis show that the method is correct and effective.