4D Chaotic System Research Articles

A Four-Dimensional No-Equilibrium Chaotic System with Multi-Scroll Chaotic Hidden Attractors and its Application in Image Encryption

Abstract In recent years, constructing hidden attractors with multi-scroll has become a key discussion point in the research and application fields of chaos science. In this paper, with the existing four-dimensional (4D) chaotic system as the base, a new four-dimensional chaotic system featuring significant characteristics of multi-scroll hidden attractors is constructed by adding a nonlinear function. Comprehensive studies including theoretical analyses and numerical simulations have been carried out on the dynamic properties of the new chaotic system, and all the results show that this system exhibits extremely complex chaotic behaviours and excellent unpredictability, which has great value in image encryption. Therefore, an image encryption scheme based on the new chaotic system is proposed, which cleverly integrates the new scrambling algorithm based on parity coordinate transformation and the new rotational diffusion algorithm. And the effectiveness of this encryption algorithm has been thoroughly analyzed and tested. The results based on the experiments show that this encryption algorithm exhibits significant advantages in performance, which can greatly enhance the security of images during encryption and transmission.

Multi-wing chaotic system based on smooth function and its predefined time synchronization

In this paper, the construction of multi-wing chaotic system and predefined time synchronization are investigated. First, a new 4D chaotic system is constructed, and the existence of chaotic attractors is proved by dissipativity, boundedness, and Lyapunov exponents. Then the existence of a saddle focus of index-2 is proved by equilibrium point analysis, and a multi-winged hyperchaotic system is generated by extending the saddle focus with the constructed smooth function, which is found to be more complex and more conducive to synchronization applications by complexity analysis. Then the simulation of the circuit also proves the chaotic properties of the system in a physical sense. Finally, a new robust controller is designed using the fast terminal sliding mode algorithm to achieve predefined time synchronization between heterogeneous chaotic systems, and the numerical simulation results indicate that this synchronization method has a rapid time of convergence and achieves the predefined time effect.

A realizable chaotic system with interesting sets of equilibria, characteristics, and its underactuated predefined-time sliding mode control

This paper presents a new 4D chaotic system having both single-scroll and double-scroll, self-excited and hidden attractors with the variation of system parameters. The proposed system exhibits several equilibrium points and their associated attractors. There are four interseting sets of equilibria with self-excited and hidden attractors. The system characteristics are demonstrated using phase plots, bifurcation diagrams and Lyapunov exponents. Offset boosting, amplitude control, the coexistence of attractors and antimonotonicity are demonstrated by the new system. Further, a predefined-time sliding mode controller with three control inputs for a 4D system is designed for successful synchronization between the two identical master and slave systems. Such an underactuated controller for synchronization is seldom seen in the literature. The developed 4D chaotic system is realized first using the NI Multisim circuit simulator and then in an NI Academic RIO hardware platform. The outcomes validate the numerical results.

Anti-control of Hopf bifurcation for a chaotic system

Abstract The anti-control of Hopf bifurcation is a method used for bifurcation control. It can be used to realize the occurrence or delay of bifurcation at the specified position to meet the needs of engineering applications. In this study, a 4D chaotic system is studied, and a dynamic state feedback control method is proposed to realize the anti-control of Hopf bifurcation for the system. By adjusting the control parameters of the controller, the system Hopf bifurcation can be generated or delayed at the specified position to realize the anti-control of Hopf bifurcation. This control method avoids complicated calculation processes and has remarkable control effects. Through numerical simulation analysis, the correctness of this control method is verified.

Image encryption scheme using a new 4-D chaotic system with a cosinoidal nonlinear term in WMSNs

To protect the sensitive data captured in the wireless media sensor networks (WMSNs), this paper propounds a novel data encryption scheme by employing a new chaotic system and the semi-tensor product compressive sensing model driven by a linear congruence generator. In this scheme, the coefficients yielded by sparsely decomposing the plaintext image onto a wavelet packet basis are firstly compressed using a key-controlled measurement matrix. To minimize the threat of various attacks, the compressed image is then subjected to bidirectional diffusion manipulation to conceal its statistical properties under the control of chaotic sequences. Additionally, it is worth explaining that a lightweight 4D-chaotic system with only one nonlinear term is designed to accommodate the resource-constrained situation in WMSNs. Finally, theoretical analysis and investigations affirm its significant enhancement in visual security, compressibility, and encryption efficiency. Moreover, the proposed scheme is significantly superior to existing relevant encryption ones.

A simple 4D no-equilibrium chaotic system with only one quadratic term and its application in pseudo-random number generator

Simple-form chaotic systems with complex behaviors have always attracted the attention of researchers. This article introduces a novel 4D chaotic system that has only a quadratic nonlinear term as compared with the existing no-equilibrium continuous chaotic ones with two quadratic or higher nonlinearity. Despite the simplicity of its model, this new system exhibits complicated chaotic behaviors. Many interesting phenomena including the coexistence of different hidden attractors, offset boosting and antimonotonicity have been discovered by various numerical analysis such as bifurcation diagrams, Lyapunov exponent spectra, and phase portraits. The further circuit simulation of this new system verifies its feasibility and practicality. Moreover, the new system is applied to construct Pseudo-Random Number Generators (PRNG), following a thorough assessment of its complexity and randomness by sample entropy. The generated numbers are subsequently assessed using 15 statistical tests. The results confirm that the new system can produce unpredictable numbers with high randomness, making it ideal for information encryption.

An image encryption algorithm using hybrid sea lion optimization and chaos theory in the hartley domain

ABSTRACT This paper implements a hybrid encryption algorithm that combines multiple chaotic systems with the Sea Lion Optimization (SLO) algorithm and Discrete Hartley Transforms. Before converting the input image to the frequency domain, it undergoes confusion using the Lorenz chaotic system and is divided into four equal parts. Once the four sections are converted into the frequency domain, the confusion operations are carried out on each section using the chaotic sequences generated by the Uruk 4D chaotic system. The SLO algorithm incorporates a third phase of confusion to enhance randomness and disrupt the statistical correlation among the image components. The ultimate image is encrypted utilizing a bit stream derived from a chaotic Nahrain 3D system. The results of security tests and analyses conducted with the new hybrid algorithm demonstrate that it exhibited a superior level of protection against predictable statistical attacks compared to conventional encryption techniques.

A No-Chatter Single-Input Finite-Time PID Sliding Mode Control Technique for Stabilization of a Class of 4D Chaotic Fractional-Order Laser Systems

Over the past decade, fractional-order laser chaotic systems have attracted a lot of attention from a variety of fields, including theoretical research as well as practical applications, which has resulted in the development of a number of different system classes. This paper introduces a novel single-input finite-time PID sliding mode control (SMC) technique to stabilize a specific group of unknown 4-dimensional chaotic fractional-order (FO) laser systems. By combining the PID concept with the FO-version of the Lyapunov stability theory, a novel finite-time PID SMC strategy has been developed, which effectively mitigates chaotic behavior in the mentioned unknown 4-dimensional chaotic FO laser system. This method makes use of a characteristic of FO chaotic systems known as boundedness, which is used here. Notably, the control input’s sign function, which is responsible for undesirable chattering, is transformed into the fractional derivative of the control input. This transformation results in a smooth and chattering-free control input, further enhancing the method’s performance. To demonstrate the efficacy of the proposed chattering-free–finite-time PID SMC technique, two numerical scenarios are presented, showcasing its efficient performance in stabilizing the unknown 4-dimensional chaotic FO laser system. These scenarios serve as illustrations of the method’s potential for practical applications.

A physical memristor-based chaotic system and its application in colour image encryption scheme

This work proposes a physical memristor (TaOx) based new 4D chaotic system with 3D multi-scroll, no equilibrium point, spiking behaviour, coexistence bursting oscillation and multistability. Using this physical memristor-based chaotic system, a novel and efficient colour image encryption algorithm has been developed using a unique box scrambling method and bit-wise XOR operations. Many interesting and new dynamics of a material-based memristive chaotic system are reported here, like 3D multi-scroll chaotic attractors, bursting characteristics, multistability, a neuronal system like spiking behaviours etc using Lyapunov spectrum and bifurcation plots. It is observed that the number of scrolls is changed with the total simulation time. This novel memristive chaotic system has limit cycles with controllable spikes and bursting oscillation. In addition, the system shows chaotic bursting oscillation under a different set of parameters and initial conditions. The coexistence of the bursting phenomena is studied here. The bursting and spiking characteristic is important for material-based memristors in neuromorphic applications. 3D Chaotic multi-scroll and multistability properties make the image encryption method more efficient and secure. Such characteristics are rare in physical memristor-based chaotic systems and using this, the image encryption algorithm is also rare in recent findings. Therefore, a new secure image encryption algorithm for colour images is proposed here, based on the unique box scrambling method, bitwise XOR operation and pseudo-random number generation using the proposed memristive chaotic system. Various tests like NPCR, UACI, histogram analysis, correlation study, information entropy analysis, robustness against external noise, etc have been performed to check the algorithm’s robustness and efficiency and test the capability to resist statistical and differential attacks.

Investigating a new conservative 4-dimensional chaotic system

This study analyzed a modified conservative 4D chaotic system is investigated using both integer and non-integer order derivatives. Several dynamical aspects of the said model are explored, such as stable equilibrium points, Lyapunov spectra (LS), attractor projection, Poincare, bifurcations and phase portrait. The system is also analyzed using a singular fractional operator, and the theory of the existence of solutions is established through functional analysis. To obtain numerical results of the fractional order system, a numerical method based on Newton polynomial is applied. The study reveals the presence of hidden and fixed point chaotic attractors for certain fractional order values.

A new encryption scheme for RGB color images by coupling 4D chaotic laser systems and the Heisenberg group

A new encryption scheme for RGB color images by coupling 4D chaotic laser systems and the Heisenberg group

A Hybrid Domain Medical Image Encryption Scheme Using URUK and WAM Chaotic Maps with Wavelet – Fourier Transforms

Image encryption is one of the most important techniques to maintain data confidentiality against illegal access and fraudulent usage. In this study, a new medical image encryption technique was developed by combining the discrete wavelet transform, the fast Fourier transform, the Arnold transform, and two multidimensional chaotic systems. The medical image is subjected to a discrete wavelet transform before the magic square shuffles the image sub-bands. Confusion operations are performed on each scrambled subdomain using the Uruk 4D chaotic system. To increase randomness and unpredictability, a second stage of confusion is implemented in the domain of the Fast Fourier transform using the Arnold transform. The final encrypted image is obtained utilizing secret keys derived from the WAM 3D chaotic system. In particular, the initial conditions for chaotic systems are derived from grayscale values, thereby increasing the algorithm’s sensitivity to the input image. The results of the tests and the security analysis indicate that the proposed algorithm is exceptionally reliable and secure.

A 4D chaotic system with four-wing attractors and hidden extreme multistability

In this paper, a novel four-wing chaotic system is constructed based on the Sprott-A system. The novel system contains three nonlinearly quadratic terms, which have rich dynamics such as four-wing hidden attractors, hidden extreme multistability and transient transitions. It is found that this system has high complexity by spectral entropy analysis. In addition, a corresponding hardware analog circuit is designed based on this system by operational amplifiers and multipliers. The experimental results agree with those of the theoretical analysis and confirm that this system is practically feasible. This system will have a good application value in secure communication and cryptography.

Block Image Encryption Algorithm Based on Novel Chaos and DNA Encoding

To solve the problem of the low secret space and security of some image schemes, a novel 4D chaotic system is derived in this paper. Compared with other similar chaotic systems, the new system only has one equilibrium point and can exhibit hyperchaotic characteristics, under some parameter space. The system has better dynamic characters represented by calculation of the Lyapunov exponents, phase planes, and visual 0–1 test diagram. In this study, a novel image encryption algorithm is employed based on the new dynamic system, Zigzag transform, and DNA operation. Based on the improved Zigzag transformation, the plain image is block-scrambled, and DNA encoded with the treated chaotic sequences. The transformation efficiency is improved by combining multiple images at the same time. Numerical analysis has been carried out; the results show that our algorithm achieves much better performance in security, i.e., with enhanced pseudo-randomness, higher key sensitivity, weak correlation, fairly large key space, higher security, and a stronger ability to resist various attacks. Through visual analysis, the algorithm is deemed safe and effective for digital images.

A New Simple Chaotic System with One Nonlinear Term

In this research article, a simple four-dimensional (4D) chaotic dynamic system with uncomplicated structure and only one nonlinear term is introduced. The features of the proposed design have been conducted with some standard nonlinear dynamic analysis and mathematical tools which show the chaotic nature. One of the most important indicators for detecting complexity of the chaotic systems is the Kaplan-York dimension of the system. Moreover, one of the main criteria of chaotic systems is its simplicity due to the reduction of operating costs. Therefore, it seems necessary to design a system as simple as possible and with high complexity. In this research, a comparison has been made between the proposed system and similar chaotic systems, which has given noticeable results. For the practical implementation of the proposed design, the circuit analysis using Multisim software has been employed. The proposed scheme has been used in the application of image encryption to show the efficiency of the proposed chaotic system and standard encryption tests have been performed. The rest of the numerical results have been conducted using MATLAB/Simulink software.

4D chaotic system-based secure data hiding method to improve robustness and embedding capacity of videos

Chaotic systems are used in various applications due to their high security and complexity. Because minor changes in parameters have a significant effect on the output values and the dynamic behavior of the system. Besides, security and robustness are the most important requirements of data hiding. Therefore, in this paper, it is proposed to use chaos-based data hiding for videos. The merits of the proposed method are selection of embedding location with robust system, increasing the secret data capacity, and keep the visual quality, which are in trade-off for existing methods. Because, in the existing methods the secret message is embedded into pixels of a consecutive, a selected or a random frame. Another deficiency of existing methods is embedding capacity. They embed fewer secret data to keep the visual quality of the stego object. However, compared to the drawback of existing literature, in the proposed method the secret message is embedded in entire video with 4D chaotic system, which is also an important novelty of this paper.The proposed method evaluated by objective criteria and achieved valuable results. The histogram analysis proves our method is undetectable. For different high ratios of embedding capacity, the visual quality of the stego videos is high where the average PSNR is about 55 dB, and MS-SSIM is closed to 1. The proposed method's results compared chaotic and non-chaotic video steganography methods. Our methods achieve 54.92 dB for embedding ratio 8.08% where the results are 53.57 dB and 5% for the best existing non-chaotic method and 49.99 dB and 0.001% for the best chaotic method, for similar capacity the proposed method gives superior visual performance over the chaotic literature. Finally, the robustness analysis with pixel value attack and lsb attack shows the proposed method appears to be resistant to these types of attacks.

A chaotic image encryption scheme based on multi-directional confusion and diffusion operations

A careful perusal of the literature regarding the image encryption algorithms indicates that many research articles based on pixel swapping operations in the horizontal and vertical orientations, have been published. This study ventures to develop a yet another sophisticated algorithm by extending the swapping operations along the diagonal–antidiagonal directions. Besides, unlike the previous studies, the confusion and diffusion operations over the pixels of the given input plain image have been crafted in a parallel fashion. This work has employed a 4D chaotic system to spawn the streams of random numbers. The initial values and the system parameters have been made dependent over the given plain image in order to inject the plaintext sensitivity in the proposed cipher. As a modus operandi of the proposed encryption algorithm, a non-intersecting pair of row and column is selected randomly with the same arbitrary length and swapped/XORed according to the value dictated by the random numbers. The same operations have been carried out for the non-intersecting pair of diagonal and anti-diagonal. The reason for selecting the non-intersecting pair stems from the fact that intersecting pairs cause to lose the precious pixels data which questions the potential invertibility feature of the proposed algorithm. To demonstrate the improved efficiency, ability to withstand the varied attacks by the cryptanalysis savvy, and prospects for some real-world application, the algorithm has been subjected to a comprehensive set of evaluation metrics.

Chaotic-DNA system for efficient image encryption

In order to prevent unwanted access to sensitive data by unauthorized individuals, color images are encoded. Because chaotic-deoxyribonucleic acid (chaotic-DNA) encoding can make information highly secure, it is often employed in image encryption. In this study, a new image encryption technique has been proposed based on a new 4D-chaotic system and DNA computing. The algorithm consists of two phases: in the first phase, the pixel positions are permuted by chaotic sequences. In the second phase, according to the concept of DNA cryptography, a set of operations (like DNA addition, DNA XOR, DNA subtraction, shift right, and shift left) are performed on the DNA encoding sequence. The performance of the suggested algorithm is evaluated through analyses like correlation coefficient, entropy, histogram, and key space. The results show that the encryption method that was exhibited has good encryption performance and high security. For encrypted images, the histogram is fairly uniform, the correlation values between adjacent pixels are very small and close to zero, and the entropy is near to the ideal value of eight. In addition, the proposed system has a very large key space that is equal to 2627 keys, which makes it resistant to brute-force, differential, and statistical attacks.

Several Control Problems of a Class of Complex Nonlinear Systems Based on UDE

This paper mainly studies several control problems of a complex 4D chaotic system. Firstly, the real part and imaginary part of the complex 4D chaotic system are separated, and the system is equivalent to a six-dimensional continuous autonomous real chaotic system. Then, the stabilization, synchronization, and anti-synchronization of the complex four-dimensional chaotic system are realized by using the control method of the combination of dynamic feedback gain control and UDE control, and the corresponding physical controllers are designed respectively. Finally, the correctness and effectiveness of the theoretical results are verified by numerical simulation.

Hidden and Self-Excited Collective Dynamics of a New Multistable Hyper-Jerk System with Unique Equilibrium

Nonlinear dynamical systems with hidden attractors belong to a recent and hot area of research. Such systems can exist in different forms, such as without equilibrium or with a stable equilibrium point. This paper focuses on the dynamics of a new 4D chaotic hyper-jerk system with a unique equilibrium point. It is shown that the new hyper-jerk system effectively exhibits different hidden behaviors, which are hidden point attractor, hidden periodic attractor, and hidden chaotic state. Collective behaviors of the system are studied in terms of the equilibrium point, bifurcation diagrams, phase portraits, frequency spectra, and two-parameter Lyapunov exponents. Some remarkable and exciting properties are found in the new snap system, such as period-doubling transition, asymmetric bubbles, and coexisting bifurcations. Also, we demonstrate that it is possible to generate different varieties of two, three, four, or five coexisting hidden and self-excited attractors in the introduced model. In addition, the amplitude and offset of the hidden chaotic attractors are perfectly controlled for possible application in engineering. Furthermore, a circuit design has been implemented to support the physical feasibility of the proposed model.