Chaotic Systems Research Articles

Image privacy protection scheme based on high-quality reconstruction DCT compression and nonlinear dynamics

The protection of digital image privacy in Big Data environment is a hot issue of increasing concern. To address the contradiction between efficiency and security for privacy protection, this paper proposes a high-quality restored image privacy protection scheme based on Discrete Cosine Transform (DCT) frequency domain compression and nonlinear dynamics. First, the RGB space of the color image is converted to YCbCr domain with 4:2:0 sampling. Then, the image is divided into sub-blocks in the spatial domain, followed by block-wise DCT and quantization into frequency domain information. Next, both Alternating Current (AC) and Direct Current (DC) coefficients of all sub-blocks are individually extracted and subjected to compression encoding. Finally, a Rubik’s Cube permutation and filtering diffusion encryption algorithm based on a nonlinear dynamical chaotic system is designed. Moreover, a mechanism for generating dynamic chaotic sequences associated with plaintext is introduced to effectively counter cryptographic attacks. The combined approach of compression and encryption significantly reduces computational complexity while improving encryption efficiency. Through experimental simulation results, the compression encryption scheme demonstrates high compression ratios as well as excellent image recovery quality while providing enhanced security against common cryptographic attacks. The work in this paper provides a preferred joint compression and encryption technical solution for digital image privacy protection in Big Data network.

Autonomous snap oscillator with only one steady state: dynamical probing, controls, pseudo-random number generation and difference synchronization

The theoretical probing, microcontroller implementation, amplitude controls, chaos control, -pseudo-random number generation (PRNG), and difference synchronization of autonomous snap oscillator with only one steady state (ASOOSS) are studied in this paper. The ASOOSS exhibits self-excited complex attractors, periodic oscillations, coexistence of chaotic hidden attractors with a stable steady state, and hidden chaotic attractors. The simulated attractors are endorsed by the microcontroller execution of ASOOSS. Then, the total and partial controls of the amplitude of ASOOSS are demonstrated by using newly inserted parameters. Moreover, the efficacy of the configured single controller in suppressing chaos within ASOOSS is demonstrated through both analytical and numerical analyses. Furthermore, the binary data generated by the ASOOSS-based PRNG successfully passes the NIST 800–22 statistical tests, providing proof of the random nature of the ASOOSS-based PRNG and making it suitable for digital applications based on chaos. Additionally, controllers are devised to enable differential synchronization of three identical coupled chaotic ASOOSS systems. The effectiveness of the differential synchronization approach is validated through numerical simulations of the coupled chaotic ASOOSS systems.

A quantum image encryption algorithm based on chaotic system and Rubik’s cube principle

A quantum image encryption algorithm based on chaotic system and Rubik’s cube principle

Adaptive fuzzy quantized prescribed performance synchronization of uncertain non-strict feedback chaotic systems with time-varying actuator failure

This paper addresses an adaptive fuzzy prescribed performance synchronization for a class of uncertain non-strict feedback chaotic systems subject to input quantization and time-varying actuator faults. The proposed approach utilizes fuzzy logic systems to estimate uncertainties. In addition, multiplicative and additive faults are considered simultaneously, which may occur either separately or simultaneously. To address the challenge, based on the backstepping scheme and a filter, an intermediate controller is conducted to compensate for the interference between actuator faults and quantification, in which a damping term and a positive time-varying function are introduced. Moreover, treating the error system as a constrained system, a simplified nonlinear mapping is employed to achieve asymmetric prescribed performance synchronization. Unlike traditional methods that rely on barrier Lyapunov functions and switch functions, the proposed approach eliminates the need for a switch function and extends the action scope to cover all synchronization errors. The analysis demonstrates that even if actuator faults and input quantization coexist, all signals in the closed-loop system remain bounded, and synchronization errors remain within the prescribed performance range. Finally, synchronization simulations are conducted on a nonlinear gyroscope system and a Non-autonomous chaotic Micro-Electro-Mechanical-System to reveal the validity of the proposed scheme.

A 4D conservative chaotic system: dynamics and realization

This paper proposes a novel four-dimensional conservative chaotic system (4D CCS) with a simple algebraic representation, comprising only two quadratic nonlinear terms. The dynamic characteristics of the 4D CCS are investigated by Poincaré mappings, Lyapunov exponents (LE), bifurcation diagrams, equilibrium points and spectral entropy (SE) complexity algorithm. Variations in parameters, initial values, and Hamiltonian energy lead to alternations between quasi-periodic and chaotic flows in the 4D CCS. The maximum Lyapunov exponent of the 4D CCS can reach a high value of 366300 under adjusting appropriate parameters and initial values. The pseudorandom sequences generated by the 4D CCS successfully pass the NIST test. Additionally, both the electronic circuit and FPGA implementation of the 4D CCS are carried out, with the experimental results aligning closely with the simulation results.

A novel color image encryption method using Fibonacci transformation and chaotic systems

INTRODUCTION: With the rapid increase in network information data, the protection of image data has become a challenging task, where image encryption technology can play an important role. This paper studies color image encryption algorithms and proposes a novel method for color image encryption to enhance the security and effectiveness of image encryption.OBJECTIVES: The purpose of this study is to effectively integrate different channel information of color images, thereby improving the effect of pixel decomposition based image encryption algorithm. Different indicators are used to analyze the effect of image encryption, and it is also compared with existing image encryption algorithms.METHODS: Initially, through pixel decomposition, the pixel values of the R, G, B channels of the color image, each with a depth of 8 bits, are decomposed into two integers between 0-15 and combined into a new data matrix. Then, multiple rounds of scrambling are performed on the transformed matrix. Next, the Fibonacci transformation matrix is applied to the scanned matrix to further change the values of its elements. Finally, XOR diffusion operation is carried out to obtain the encrypted image.RESULTS: Experimental results show that the proposed method achieves relatively good results in multiple image encryption indicator tests. The algorithm not only inherits the advantages of existing image encryption but also effectively integrates the information of each channel of the color image, providing better security.CONCLUSION: This study further proves the effectiveness of image encryption algorithms based on pixel decomposition and provides a new idea for better color image encryption algorithms, which is expected to be applied to other issues such as information hiding and data protection.

Mixmaster chaos in an AdS black hole interior

We derive gravitational backgrounds that are asymptotically Anti-de Sitter, have a regular black hole horizon and which deep in the interior exhibit mixmaster chaotic dynamics. The solutions are obtained by coupling gravity with a negative cosmological constant to three massive vector fields, within an Ansatz that reduces to ordinary differential equations. At late interior times the equations are identical to those analysed in depth by Misner and by Belinskii-Khalatnikov-Lifshitz fifty years ago. We review and extend known classical and semiclassical results on the interior chaos, formulated as both a dynamical system of ‘Kasner eras’ and as a hyperbolic billiards problem. The volume of the universe collapses doubly-exponentially over each Kasner era. A remarkable feature is the emergence of a conserved energy, and hence a ‘time-independent’ Hamiltonian, at asymptotically late interior times. A quantisation of this Hamiltonian exhibits arithmetic chaos associated with the principal congruence subgroup Γ(2) of the modular group. We compute a large number of eigenvalues numerically to obtain the spectral form factor. While the spectral statistics is anomalous for a chaotic system, the eigenfunctions themselves display random matrix behaviour.

A hybrid model of recursive cellular automata, DNA sequences, and chaotic system for image encryption

A hybrid model of recursive cellular automata, DNA sequences, and chaotic system for image encryption

A fixed point analysis of fractional dynamics of heat transfer in chaotic fluid layers

This manuscript aims to analyze the chaotic fluid layer heating from below, using the Atangana–Baleanu derivative to characterize the existence and uniqueness of this chaotic model based on the provided fixed point data. First, with suitable examples, we demonstrate the existence and uniqueness of fixed point solutions within the framework of controlled metric spaces. Secondly, we study the chaotic behavior of water with various Rayleigh numbers by experimenting with it while keeping a constant Prandtl number. Furthermore, the role of the Rayleigh number in magnifying chaotic behaviors is examined, with graphical simulations demonstrating the effect of different fractional orders of the Atangana–Baleanu derivative. This research provides insights into the dynamics of chaotic systems and their practical applications in controlled situations.

Autonomous three-dimensional oscillator with two and four wings attractors embedded in the microcontroller: analysis, amplitude controls, random number generator, and image encryption application

Robust chaotic systems offer unpredictability, complex dynamics, noise-like properties, efficient bifurcation behavior, and the ability to model real-world phenomena, making them valuable in diverse scientific and engineering applications. This paper details on the dynamical appraisal, amplitude controls, microcontroller execution, Random number generator (RNG) of an autonomous three-dimensional (3D) oscillator with two and four wings attractors (ATDOTFWA), and its image encryption application. Thanks to the Routh-Hurwitz criteria, five steady states found in the ATDOTFWA are classified as stable or unstable, depending on its two control parameters. During the numerical simulations employing the Runge–Kutta scheme, the ATDOTFWA exhibit a wide range of dynamic behaviors, including no oscillations, Hopf bifurcation, limit cycle, five distinct presentations of two wings chaotic structures, monostable and bistable two wings chaotic structures, bistable and monostable regular oscillations, chaotic bursting characteristics, coexistence of period-2-oscillations and four wings chaotic structure, and four wings chaotic attractor which were validated experimentally by the microcontroller implementation. The total and partial controls of the amplitude are achieved in the ATDOTFWA. A RNG is designed based on the ATDOTFWA, and the generated random numbers are successfully tested using the ENT and NIST 800–22 statistical test suites, demonstrating the reliability of the ATDOTFWA-based RNG. This reliability is further confirmed through the application of the ATDOTFWA-based RNG in an efficient and secure image encryption process, where the generated random numbers are used as the encryption key. The effectiveness of the image encryption process is validated through comprehensive cryptanalysis, with an encryption time of 0.1923 s for a 512×512 image, an average normalized pixel change rate (NPCR) of 99.6126%, an average unified average changing intensity (UACI) of 33.4578%, and an average information entropy of 7.9994.

Circuit realization and application of a chaotic system with hidden attractor, controlled spike discharge and offset boosting

Circuit realization and application of a chaotic system with hidden attractor, controlled spike discharge and offset boosting

Industrial Robot Trajectory Optimization Based on Improved Sparrow Search Algorithm

This paper proposes an enhanced multi-strategy sparrow search algorithm to optimize the trajectory of a six-axis industrial robot, addressing issues of low efficiency and high vibration impact on joints during operation. Initially, the improved D-H parametric method is employed to establish both forward and inverse kinematic models of the robot. Subsequently, a 3-5-3 mixed polynomial interpolation trajectory planning approach is applied to the robot. Building upon the conventional sparrow algorithm, a two-dimensional Logistic chaotic system initializes the population. Additionally, a Levy flight strategy and nonlinear adaptive weighting are introduced to refine the discoverer position update operator, while an inverse learning strategy enhances the vigilante position update operator. These modifications boost both the local and global search capabilities of the algorithm. The improved sparrow algorithm, based on 3-5-3 hybrid polynomial trajectory planning, is then used for the time-optimal trajectory planning of the robot. This is compared with traditional sparrow search algorithm and particle swarm algorithm optimization results. The findings indicate that the proposed enhanced sparrow search algorithm outperforms both the standard sparrow algorithm and the particle swarm algorithm in terms of convergence speed and accuracy for robot trajectory optimization. This can lead to the increased work efficiency and performance of the robot.

Chaos Control and Synchronization of a New Fractional Laser Chaotic System

In this article, we introduce a new fractional laser chaotic system derived from the Lorenz–Haken equations. We investigate the complex dynamics of the proposed system consisting chaos, stability, control and synchronization of chaos. Moreover, we numerically reveal the nonlinear dynamics of the fractional laser chaotic system through the phase portraits, time histories and bifurcation diagrams. Also, we indicate the chaotic behaviors of the system by means of Lyapunov exponents, bifurcation diagrams versus all parameters along the state variables, phase portraits and time histories with different trajectories and initial conditions. The necessary conditions to eliminate the chaotic vibration of the system are obtained via the feedback control procedure. Meanwhile, a synchronization mechanism based on the feedback control technique is achieved for coupled fractional laser chaotic systems. Furthermore, we show that the fractional derivative order is very effective on reducing the irregular and chaotic behaviors of the system.

Bifurcation analysis with chaotic attractor for a special case of jerk system

This article focuses on investigating local bifurcations in a special type of chaotic jerk system. It examines the occurrence and non-occurrence of saddle-node, transcritical, zero-Hopf, Hopf, and pitchfork bifurcations at the origin. The parameters that result in a zero-Hopf equilibrium point at the origin are characterized for the proposed system. Additionally, a demonstration is provided to show that the utilization of the first-order averaging theory leads to the emergence of a single periodic solution branching out from the zero-Hopf equilibrium located at the origin. Furthermore, the focus quantities method is applied to explore the periodicity of the cubic part of the system. This method helps determine the number of periodic solutions that can emerge from the Hopf point. Due to the computational load for computing singular quantities, only three singular quantities are found. Under specific conditions, it is shown that three periodic solutions can bifurcate from the origin of the system. Finally, the study also examines the chaotic attractors of the system.

A spatiotemporal chaos based deep learning model watermarking scheme

With deep learning techniques achieving great results in modern industry, the intellectual property (IP) protection for deep learning models has attracted the attention of academics and engineers. However, training a commercially viable deep learning model usually needs professional resources and time. Once a malicious user clones, illegally distributes and uses the model, it can infringe on the model owner's IP and even steal its market share. Among the existing IP protection methods, scholars prefer the black-box watermarking approaches, of which the content of the trigger set and the label are the key part of the watermarking technique. However, most schemes do not consider the security and invisibility of the trigger set, which allows attackers to easily trigger the model by creating a fake trigger set, thereby committing a fraudulent ownership claim attack and claiming the ownership belongs to themselves. To overcome these drawbacks, we proposed a spatiotemporal chaotic data annotation method. Firstly, the unpredictability and acyclicity of chaos make the model resistant to fraudulent ownership claim attacks, statistical inference and other common machine learning attacks; Secondly, the trigger set and parameters are independent of each other, guaranteeing the security of the key; Thirdly, the spatiotemporal chaotic system provides a large key space, which meets the commercialization needs of deep learning models. Theoretical analysis and experimental results show that our scheme has security, practicality and robustness. To further validate the superiority of the proposed method, we also compare it with the Logistic chaotic annotation watermarking-based method, and the results show that our method performs better in terms of robustness, effectiveness, completeness, fidelity, security and practicality.

A Two-Dimensional Discrete Memristor Map: Analysis and Implementation

In this paper, we present a novel two-dimensional discrete memristor map that is based on a discrete memristor model and a sine–arcsine one-dimensional map. First, an analysis is conducted on the memristor model to understand its characteristics. Then, the model is coupled with the sine–arcsine one-dimensional map to achieve the two-dimensional discrete memristor map. Our investigation reveals the presence of coexisting attractors and hyperchaotic attractors as the bifurcation parameters vary. Numerical simulations show that the discrete memristors effectively enhance the complexity of chaos in the sine–arcsine map. Furthermore, a digital circuit is designed to experimentally verify the new chaotic system. The research results can enrich the theoretical analysis and circuit implementation of chaos.

Design and Analysis of a Novel Fractional-Order System with Hidden Dynamics, Hyperchaotic Behavior and Multi-Scroll Attractors

The design of chaotic systems with complex dynamic behaviors has always been a key aspect of chaos theory in engineering applications. This study introduces a novel fractional-order system characterized by hidden dynamics, hyperchaotic behavior, and multi-scroll attractors. By employing fractional calculus, the system’s order is extended beyond integer values, providing a richer dynamic behavior. The system’s hidden dynamics are revealed through detailed numerical simulations and theoretical analysis, demonstrating complex attractors and bifurcations. The hyperchaotic nature of the system is verified through Lyapunov exponents and phase portraits, showing multiple positive exponents that indicate a higher degree of unpredictability and complexity. Additionally, the system’s multi-scroll attractors are analyzed, showcasing their potential for secure communication and encryption applications. The fractional-order approach enhances the system’s flexibility and adaptability, making it suitable for a wide range of practical uses, including secure data transmission, image encryption, and complex signal processing. Finally, based on the proposed fractional-order system, we designed a simple and efficient medical image encryption scheme and analyzed its security performance. Experimental results validate the theoretical findings, confirming the system’s robustness and effectiveness in generating complex chaotic behaviors.

Application of Fractional-Order Multi-Wing Chaotic System to Weak Signal Detection

This work investigates a fractional-order multi-wing chaotic system for detecting weak signals. The influence of the order of fractional calculus on chaotic systems’ dynamical behavior is examined using phase diagrams, bifurcation diagrams, and SE complexity diagrams. Then, the principles and methods for determining the frequencies and amplitudes of weak signals are examined utilizing fractional-order multi-wing chaotic systems. The findings indicate that the lowest order at which this kind of fractional-order multi-wing chaotic system appears chaotic is 2.625 at a=4, b=8, and c=1, and that this value decreases as the driving force increases. The four-wing and double-wing change dynamics phenomenon will manifest in a fractional-order chaotic system when the order exceeds the lowest order. This phenomenon can be utilized to detect weak signal amplitudes and frequencies because the system parameters control it. A detection array is built to determine the amplitude using the noise-resistant properties of both four-wing and double-wing chaotic states. Deep learning images are then used to identify the change in the array’s wing count, which can be used to determine the test signal’s amplitude. When frequencies detection is required, the MUSIC method estimates the frequencies using chaotic synchronization to transform the weak signal’s frequencies to the synchronization error’s frequencies. This solution adds to the contact between fractional-order calculus and chaos theory. It offers suggestions for practically implementing the chaotic weak signal detection theory in conjunction with deep learning.

A Secure Image Encryption Scheme Based on a New Hyperchaotic System and 2D Compressed Sensing.

In insecure communication environments where the communication bandwidth is limited, important image data must be compressed and encrypted for transmission. However, existing image compression and encryption algorithms suffer from poor image reconstruction quality and insufficient image encryption security. To address these problems, this paper proposes an image-compression and encryption scheme based on a newly designed hyperchaotic system and two-dimensional compressed sensing (2DCS) technique. In this paper, the chaotic performance of this hyperchaotic system is verified by bifurcation diagrams, Lyapunov diagrams, approximate entropy, and permutation entropy, which have certain advantages over the traditional 2D chaotic system. The new 2D chaotic system as a pseudo-random number generator can completely pass all the test items of NIST. Meanwhile, this paper improves on the existing 2D projected gradient (2DPG) algorithm, which improves the quality of image compression and reconstruction, and can effectively reduce the transmission pressure of image data confidential communication. In addition, a new image encryption algorithm is designed for the new 2D chaotic system, and the security of the algorithm is verified by experiments such as key space size analysis and encrypted image information entropy.

Color Image Encryption Based on an Evolutionary Codebook and Chaotic Systems.

Encryption of images is an important method that can effectively improve the security and privacy of crucial image data. Existing methods generally encrypt an image with a combination of scrambling and encoding operations. Currently, many applications require highly secure results for image encryption. New methods that can achieve improved randomness for both the scrambling and encoding processes in encryption are thus needed to further enhance the security of a cipher image. This paper proposes a new method that can securely encrypt color images. As the first step of the proposed method, a complete bit-level operation is utilized to scramble the binary bits in a color image to a full extent. For the second step, the bits in the scrambled image are processed with a sweeping operation to improve the encryption security. In the final step of encryption, a codebook that varies with evolutionary operations based on several chaotic systems is utilized to encrypt the partially encrypted image obtained in the second step. Experimental results on benchmark color images suggest that this new approach can securely encrypt color images and generate cipher images that remain secure under different types of attacks. The proposed approach is compared with several other state-of-the-art encryption approaches and the results show that it can achieve improved encryption security for cipher images. Experimental results thus suggest that this new approach can possibly be utilized practically in applications where color images need to be encrypted for content protection.