Dynamical Analysis, Chaos Synchronization, and Image Encryption Application of a Novel Variable-Order Fractal-Fractional Memristor-Based Hyperchaotic System

Memristor-based systems and their potential applications, in which memristor is both a nonlinear element and a memory element, have been received significant attention in the control literature. In this work, we study a memristor-based hyperchaotic system with hidden attractors. First, we study the dynamic properties of the memristor-based hyperchaotic system such as equilibria, Lyapunov exponents, Kaplan-Yorke dimension, etc. We obtain the Lyapunov exponents of the memristor-based system as \(L_1 = 0.2205\), \(L_2 = 0.0305\), \(L_3 = 0\) and \(L_4 = -10.7862\). Since there are two positive Lyapunov exponents, the memristor-based system is hyperchaotic. Also, the Kaplan-Yorke fractional dimension of the memristor-based hyperchaotic system is obtained as \(D_{KY} = 3.0233\), which shows the high complexity of the system. We show that the memristor-based hyperchaotic system has no equilibrium point, which shows that the system has a hidden attractor. Control and synchronization of chaotic and hyperchaotic systems are important research problems in chaos theory. Sliding mode control is an important method used to solve various problems in control systems engineering. In robust control systems, the sliding mode control is often adopted due to its inherent advantages of easy realization, fast response and good transient performance as well as insensitivity to parameter uncertainties and disturbance. Next, using integral sliding mode control, we design adaptive control and synchronization schemes for the memristor-based hyperchaotic system. The main adaptive control and synchronization results are established using Lyapunov stability theory. MATLAB simulations are shown to illustrate all the main results of this work.

Memristor-based systems and their potential applications, in which memristor is both a nonlinear element and a memory element, have been received significant attention in the control literature. In this work, we study a memristor-based hyperchaotic system with hidden attractors. First, we study the dynamic properties of the memristor-based hyperchaotic system such as equilibria, Lyapunov exponents, Poincare map, etc. We obtain the Lyapunov exponents of the memristor-based system as \(L_1 = 0.1244\), \(L_2 = 0.0136\), \(L_3 = 0\) and \(L_4 = -10.8161\). Since there are two positive Lyapunov exponents, the memristor-based system is hyperchaotic. Also, the Kaplan-Yorke fractional dimension of the memristor-based hyperchaotic system is obtained as \(D_{KY} = 3.0128\). Next, we design adaptive control and synchronization schemes for the memristor-based hyperchaotic system. The main adaptive control and synchronization results are established using Lyapunov stability theory. MATLAB simulations are shown to illustrate all the main results of this work. Finally, an electronic circuit emulating the memristor-based hyperchaotic system has been designed using off-the-shelf components.

Memristor-based systems and their potential applications, in which memristor is both a nonlinear element and a memory element, have been received significant attention in the control literature. In this work, we propose a conservative memristor-based hyperchaotic hyperjerk system with infinite number of equilibrium points. In classical mechanics, the third-order time-derivative of displacement is called jerk, while the fourth-order time-derivative of displacement is called snap. As a result, a dynamical system which is represented by an nth order ordinary differential equation with \(n > 3\) is considered as a hyperjerk system. Hyperjerk systems have received significant attention in the control literature. In this research work, a conservative memristor-based hyperjerk system has been designed which displays rich, hyperchaotic behavior. Interestingly, this hyperjerk system displays an infinite number of equilibrium points because of the presence of a memristive device. In this work, we obtain the Lyapunov exponents of the memristor-based system as \(L_1 = 0.2098\), \(L_2 = 0.2035\), \(L_3 = 0\) and \(L_4 = -0.4133\). Since there are two positive Lyapunov exponents, the memristor-based system is hyperchaotic. Also, the Kaplan-Yorke dimension of the memristor-based hyperchaotic system is obtained as \(D_{KY} = 4\). Next, we design adaptive control and synchronization schemes for the memristor-based hyperjerk system with unknown parameters via backstepping control method. The main adaptive control and synchronization results are established using Lyapunov stability theory. MATLAB simulations are shown to illustrate all the main results of this work.

A three-dimensional autonomous deterministic chaotic system having six parameters is explored within this article. The dynamical characteristics of the proposed system are investigated through eigenvalues structure, bifurcation diagrams, Kaplan–Yorke dimension, Lyapunov exponents, time response, and phase plane trajectories. For the suitable design of the system parameters, it is found that the system can exhibit periodic, period-n, or chaotic oscillations. Accordingly, the system’s dynamical behavior to the variation of its coefficients has been explored. The obtained results revealed that the proposed dynamical system does not lose its chaotic oscillations for the small fluctuations of one or more of the values of its parameters. In addition, chaos control and chaos synchronization have been studied by means of the adaptive control strategy relying on Lyapunov’s second method of stability. The numerical simulation revealed that superior chaos control and master-slave synchronization have been achieved by the applied control laws. Finally, the obtained results have been simulated via a nonlinear electronic circuit that demonstrated the feasibility of the purposed chaotic system for different engineering applications such as secure communications, cryptosystems, image encryption, and image processing.

This paper presents a novel four-dimensional (4D) memristive system, notable for its simplicity and unique dynamic behaviors. Comprising only seven terms and devoid of equilibrium points, this system is capable of generating hidden attractors. A comprehensive analysis of its dynamic properties is conducted, including Lyapunov exponents, Kaplan–Yorke dimensions, temporal diagrams and phase portraits, multistability, and offset boosting. Numerical simulations over a sufficiently long time interval show that the Lyapunov function remains bounded, thereby confirming the dissipative nature and global stability of the newly proposed 4D hyperchaotic system. The theoretical model is further validated through electronic simulations of the chaotic system using Multisim software. Additionally, the paper explores synchronization between two identical 4D hyperchaotic systems. The proposed system, despite its structural simplicity, exhibits intricate chaotic dynamics, making it suitable for various practical applications. In particular, a method for chaotic encryption and decryption of an information signal was developed and validated through numerical testing. Based on the method of complete synchronization of chaotic systems, the possibility of detecting a weak signal is demonstrated. The proposed system is also implemented on an Arduino UNO to demonstrate its practical applicability for real-time chaotic signal generation and image encryption.

A memristor-based VB2 chaotic system: Dynamical analysis, circuit implementation, and image encryption

Memristors exhibit controllable nonlinear characteristics, generating chaotic signals that are characterized by randomness, sensitivity, and unpredictability, thereby demonstrating significant potential applications in information encryption and signal processing. With the integration of chaos theory and electronic technology, constructing memristive hyperchaotic systems has become a hot topic in nonlinear science and information security. To overcome the limitation of monotonic dynamic characteristics in traditional chaotic systems, we design a novel memristor-based hyperchaotic system with richer dynamic behavior and higher application value in this paper. Moreover, the characteristic analysis, theoretical verification, application exploration, and hardware implementation are conducted to support the engineering applications of the system. Building upon the classical Chen system, this work is innovatively combined with a cubic nonlinear magnetically controlled memristor model as a feedback element. By establishing a mathematical model of the memristor and coupling it with the state equations of the Chen system, we design a four-dimensional memristor-based hyperchaotic system. First, by integrating numerical computation with differential equation theory, a comprehensive mathematical model is established to analyze fundamental properties, such as symmetry and dissipativity, thereby validating the system’s rationality. Second, the system’s dynamical behaviors are analyzed, including attractor phase diagrams, Lyapunov exponents, power spectra, parameter effects, transient dynamics, and coexisting attractors. Simultaneously, variational methods are utilized to analyze unstable periodic orbits within the system. A symbolic coding approach based on orbital characteristics is established to convert orbital information into symbolic sequences, and orbital pruning rules are explored to provide a basis for optimal orbital control. Furthermore, a digital image encryption method is proposed based on this system. Using chaotic sequences as keys, image pixels are scrambled and diffused. The effectiveness of encryption is validated through histogram analysis, correlation analysis, information entropy evaluation, and testing of anti-attack capabilities. Finally, a DSP-based digital circuit hardware platform is constructed to run the system, and the hardware experimental results are compared with software simulation outcomes. These findings reveal that the introduction of memristors induces linearly distributed equilibrium points in phase space, generating hidden attractors that enrich the chaotic behavior of the system. The simulation of dynamic behavior confirms the rich dynamics of this four-dimensional memristor-based hyperchaotic system. The proposed digital image encryption method demonstrates robust security performance. The DSP hardware experiments and software simulations yield highly consistent attractor phase diagrams, validating the correctness and feasibility of the system.

This paper proposes a novel hybrid image encryption scheme that combines a memristor-based hyperchaotic system, the Trivium stream cipher, DNA-inspired operations, and a lightweight 3DES layer into a tightly integrated multi-layer architecture. The memristor-based chaotic system ensures a large key space and complex dynamics, while the Trivium cipher enhances randomness, and the DNA operations introduce strong confusion and diffusion properties. The addition of 3DES further reinforces resistance against brute-force and structural attacks. Extensive experiments demonstrate the effectiveness of the proposed algorithm: the information entropy of encrypted images approaches the ideal value (7.999), the correlation coefficients between adjacent pixels are reduced to nearly zero (≈ 0.001), and differential attack metrics achieve NPCR values above 99.58% and UACI values above 33%. Furthermore, the scheme exhibits robustness against data loss and noise (up to 20% salt-and-pepper noise) with minimal degradation in decrypted images. Hardware implementation on the NVIDIA Jetson TX2 confirms the practicality of the method for embedded and edge-computing environments. These results indicate that the proposed scheme provides a secure, efficient, and hardware-friendly solution for modern image encryption applications.

Initial value-related dynamical analysis of the memristor-based system with reduced dimensions and its chaotic synchronization via adaptive sliding mode control method

Memristor, as a typical nonlinear element, is able to produce chaotic signals in chaotic systems easily. Chaotic systems have potential applications in secure communications, information encryption, and other fields. Therefore, it is of importance to generate abundant dynamic behaviors in a single chaotic system. In this paper, a novel memristor-based chaotic system without equilibrium points is proposed. One of the essential features is the absence of symmetry in this system, which increases the complexity of the new system. Then, the nonlinear dynamic behaviors of the system are analyzed in terms of chaos diagrams, bifurcation diagrams, Poincaré maps, Lyapunov exponent spectra, the sum of Lyapunov exponents, phase portraits, 0–1 test, recurrence analysis and instantaneous phase. The results of the sum of Lyapunov exponents show that the given system is a quasi-Hamiltonian system with certain initial conditions (IC) and parameters. Next, other critical phenomena, such as hidden multi-scroll attractors, abundant coexistence characteristics, are found characterized through basins of attraction and others. Especially, it reveals some rare phenomena in other systems that multiple hidden hyperchaotic attractors coexist. Finally, the circuit implementation based on Micro Control Unit (MCU) confirms theoretical analysis and the numerical simulation.

This paper proposes a simple no-equilibrium chaotic system with only one signum function as compared with the existing no-equilibrium chaotic ones with at least one quadratic or higher nonlinearity. The system has the offset boosting of three variables through adjusting the corresponding controlled constants. The resulting hidden attractors can be distributed in a 1D line, a 2D lattice, a 3D grid, and even in an arbitrary location of the phase space. Particularly, a hidden chaotic bursting oscillation is also observed in this system, which is an uncommon phenomenon. In addition, complex hidden dynamics is investigated via phase portraits, time series, Kaplan-Yorke dimensions, bifurcation diagrams, Lyapunov exponents, and two-parameter bifurcation diagrams. Then, a very simple hardware circuit without any multiplier is fabricated, and the experimental results are presented to demonstrate theoretical analyses and numerical simulations. Furthermore, the randomness test of the chaotic pseudo-random sequence generated by the system is tested by the National Institute of Standards and Technology test suite. The tested results show that the proposed system has good randomness, thus being suitable for chaos-based applications such as secure communication and image encryption.

The widespread use of information and communication tools today facilitates information access and highlights the significance of information and data security. In recent years, chaos-based encryption systems have emerged as a promising approach for protecting the confidentiality of transmitted images. In particular, memristor-based hyperchaotic systems have attracted significant attention because of their robustness and complexity. In this paper, we propose an image encryption model that employs a two-stage encryption method using various chaotic systems, including the logistic map, Lorenz chaotic system, and memristor-based hyperchaotic system, with AES and RSA encryption algorithms. The proposed hybrid scheme applies bit-based pixel diffusion and confusion techniques to improve the security of encrypted images. Statistical and security tests are conducted to compare the performance of the different encryption systems and algorithms and to present the measurement values obtained from the analysis. Our experimental results demonstrate the effectiveness of the proposed image encryption scheme in terms of security, speed, and reliability and provide valuable insights for the development of future chaos-based encryption systems.

With the advancement of technology, the growth of multimedia and communication tools has sped up data transfer, and guaranteeing image security has become a crucial concern, particularly during the transmission and storage of images. So, when images are sent over a public network, they should be encrypted before being sent to the receiving part. In this study, a memristor-based encryption system with Deoxyribonucleic acid (DNA) coding is proposed on the LabVIEW platform to ensure information security. Firstly, memrsitor based hyperchaotic system is used for chaotic sequence. The images are encrypted using the DNA and XOR arithmetic process on the LabVIEW platform. A memristor-based hyperchaotic system and the combination of techniques used are aimed at encrypting the image securely. In addition, security tests; histogram analysis, correlation analysis, differential attack, and entropy analysis, are performed on the proposed system and the results are presented. The aforementioned methods are thoroughly examined and tested to determine their efficacy. It has been determined that the proposed encryption schemes are effective and can therefore be used in real-time applications.

Chapter 9 - Memristor-based novel 4D chaotic system without equilibria: Analysis and projective synchronization

This paper presents a new class of chaotic systems with infinite number of equilibrium points like a three-leaved clover. They signify an exciting class of dynamical systems which represent many major characteristics of regular and chaotic motions. These chaotic systems belong to the general class of chaotic systems with hidden attractors. By using a systematic computer search, three chaotic systems with three-leaved-clover-shaped equilibria were found which are classified into dissipative systems. Dynamics of the chaotic system with the three-leaved-clover-equilibria has been investigated by using phase portraits, bifurcation diagram, Lyapunov exponents, Kaplan–Yorke dimension and Poincare map. Moreover, an electronic circuit implementation of the theoretical system is designed to check its effectiveness. Random number generator design has been realized with newly developed chaotic systems. The obtained random bit sequences are used for image encryption. Security analysis of image encryption processes has been performed.