A robust color image encryption scheme for optical applications using a novel zigzag technique and 3D fractional-order chaotic system

Shilnikov criteria believe that the emergence of chaos requires at least one unstable equilibrium, and an attractor is associated with the unstable equilibrium. However, some special chaotic systems have been proposed recently, each of which has one stable equilibrium, or no equilibrium at all, or has a linear equilibrium (infinite equilibrium). These special dynamical systems can present chaotic characteristics, and the attractors in these chaotic systems are called hidden attractors due to the fact that the attraction basins of chaotic systems do not intersect with small neighborhoods of any equilibrium points. Since they were first found and reported in 2011, the dynamical systems with hidden attractors have attracted much attention. Additionally, the fractional-order system, which can give a clearer physical meaning and a more accurate description of the physical phenomenon, has been broadly investigated in recent years. Motivated by these two considerations, in this paper, we propose a fractional-order chaotic system with hidden attractors, and the finite time synchronization of the fractional-order chaotic systems is also studied.Most of the researches mainly focus on dynamic analysis and control of integer-order chaotic systems with hidden attractors. In this paper, based on the Sprott E system, a fractional-order chaotic system is constructed by adding an appropriate constant term. The fractional-order chaotic system has only one stable equilibrium point, but it can generate various hidden attractors. Basic dynamical characteristics of the system are analyzed carefully through phase diagram, Poincare mapping and power spectrum, and the results show that the fractional-order system can present obvious chaotic characteristics. Based on bifurcation diagram of system order, it can be found that the fractional-order system can have period attractors, doubling period attractors, and chaotic attractors with various orders. Additionally, a finite time synchronization of the fractional-order chaotic system with hidden attractors is realized based on the finite time stable theorem, and the proposed controller is robust and can guarantee fast convergence. Finally, numerical simulation is carried out and the results verify the effectiveness of the proposed controller.The fractional-order chaotic system with hidden attractors has more complex and richer dynamic characteristics than integer-order chaotic systems, and chaotic range of parameters is more flexible, meanwhile the dynamics is more sensitive to system parameters. Therefore, the fractional-order chaotic system with hidden attractors can provide more key parameters and present better performance for practical applications, such as secure communication and image encryption, and it deserves to be further investigated.

Fractional-order chaotic systems have more complex dynamics than integer-order chaotic systems. Thus, investigating fractional chaotic systems for the creation of image cryptosystems has been popular recently. In this article, a fractional-order memristor has been developed, tested, numerically analyzed, electronically realized, and digitally implemented. Consequently, a novel simple three-dimensional (3D) fractional-order memristive chaotic system with a single unstable equilibrium point is proposed based on this memristor. This fractional-order memristor is connected in parallel with a parallel capacitor and inductor for constructing the novel fractional-order memristive chaotic system. The system’s nonlinear dynamic characteristics have been studied both analytically and numerically. To demonstrate the chaos behavior in this new system, various methods such as equilibrium points, phase portraits of chaotic attractor, bifurcation diagrams, and Lyapunov exponent are investigated. Furthermore, the proposed fractional-order memristive chaotic system was implemented using a microcontroller (Arduino Due) to demonstrate its digital applicability in real-world applications. Then, in the application field of these systems, based on the chaotic behavior of the memristive model, an encryption approach is applied for grayscale original image encryption. To increase the encryption algorithm pirate anti-attack robustness, every pixel value is included in the secret key. The state variable’s initial conditions, the parameters, and the fractional-order derivative values of the memristive chaotic system are used for contracting the keyspace of that applied cryptosystem. In order to prove the security strength of the employed encryption approach, the cryptanalysis metric tests are shown in detail through histogram analysis, keyspace analysis, key sensitivity, correlation coefficients, entropy analysis, time efficiency analysis, and comparisons with the same fieldwork. Finally, images with different sizes have been encrypted and decrypted, in order to verify the capability of the employed encryption approach for encrypting different sizes of images. The common cryptanalysis metrics values are obtained as keyspace = 2648, NPCR = 0.99866, UACI = 0.49963, H(s) = 7.9993, and time efficiency = 0.3 s. The obtained numerical simulation results and the security metrics investigations demonstrate the accuracy, high-level security, and time efficiency of the used cryptosystem which exhibits high robustness against different types of pirate attacks.

Fractional-order double-ring erbium-doped fiber laser chaotic system and its application on image encryption

In order to figure out the dynamical behaviour of a fractional-order chaotic system and its relation to an integer-order chaotic system, in this paper we investigate the synchronization between a class of fractional-order chaotic systems and integer-order chaotic systems via sliding mode control method. Stability analysis is performed for the proposed method based on stability theorems in the fractional calculus. Moreover, three typical examples are carried out to show that the synchronization between fractional-order chaotic systems and integer-orders chaotic systems can be achieved. Our theoretical findings are supported by numerical simulation results. Finally, results from numerical computations and theoretical analysis are demonstrated to be a perfect bridge between fractional-order chaotic systems and integer-order chaotic systems.

Aiming at the complexity problem of fractional-order Jafari-Sprott chaotic system, in this paper, Adomian decomposition method is used to study its numerical analysis and a complexity analysis method of fractional-order Jafari-Sprott chaotic system based on fuzzy entropy algorithm, sample entropy algorithm and dispersion entropy algorithm is proposed. For the synchronization and control of fractional-order Jafari-Sprott chaotic system, sliding mode control is used to achieve synchronization of fractional-order Jafari-Sprott chaotic system and a control method of fractional-order Jafari-Sprott chaotic system is proposed based on frequency distribution model of fractional-order integral operator. The main results are as follows: (1) The complexity of the fractional-order Jafari-Sprott chaotic system is greater than the integer-order Jafari-Sprott chaotic system, and fractional-order chaotic system has better application prospects. (2) Moreover, it is concluded that the effect of the dispersion entropy algorithm on detecting complexity is the best, which provides theoretical and experimental basis for the practical engineering application of the fractional-order Jafari-Sprott chaotic system. (3) Synchronization and control of fractional-order Jafari-Sprott chaotic system is accomplished by sliding model control and frequency distribution model of fractional-order integral operator respectively. In particular, the control effect of each variable is accomplished by designing a control law based on the frequency distribution model of fractional integral operator.

Dynamic analysis of an improper fractional-order laser chaotic system and its image encryption application

Chaos-based image encryption has become a prominent area of research in recent years. In comparison to ordinary chaotic systems, fractional-order chaotic systems tend to have a greater number of control parameters and more complex dynamical characteristics. Thus, an increasing number of researchers are introducing fractional-order chaotic systems to enhance the security of chaos-based image encryption. However, their suggested algorithms still suffer from some security, practicality, and efficiency problems. To address these problems, we first constructed a new fractional-order 3D Lorenz chaotic system and a 2D sinusoidally constrained polynomial hyper-chaotic map (2D-SCPM). Then, we elaborately developed a multi-image encryption algorithm based on the new fractional-order 3D Lorenz chaotic system and 2D-SCPM (MIEA-FCSM). The introduction of the fractional-order 3D Lorenz chaotic system with the fourth parameter not only enables MIEA-FCSM to have a significantly large key space but also enhances its overall security. Compared with recent alternatives, the structure of 2D-SCPM is simpler and more conducive to application implementation. In our proposed MIEA-FCSM, multi-channel fusion initially reduces the number of pixels to one-sixth of the original. Next, after two rounds of plaintext-related chaotic random substitution, dynamic diffusion, and fast scrambling, the fused 2D pixel matrix is eventually encrypted into the ciphertext one. According to numerous experiments and analyses, MIEA-FCSM obtained excellent scores for key space (2541), correlation coefficients (<0.004), information entropy (7.9994), NPCR (99.6098%), and UACI (33.4659%). Significantly, MIEA-FCSM also attained an average encryption rate as high as 168.5608 Mbps. Due to the superiority of the new fractional-order chaotic system, 2D-SCPM, and targeted designs, MIEA-FCSM outperforms many recently reported leading image encryption algorithms.

This paper investigates the function projective synchronization between fractional-order chaotic systems and integer-order chaotic systems using the stability theory of fractional-order systems. The function projective synchronization between three-dimensional (3D) integer-order Lorenz chaotic system and 3D fractional-order Chen chaotic system are presented to demonstrate the effectiveness of the proposed scheme.

Synchronization between a fractional-order system and an integer order system

In this paper, we bring attention to synchronization between a fractional-order chaotic system and an integer order chaotic system, which is very challenging because it can form a bridge between a fractional-order chaotic system and an integer order chaotic system. More specifically, we present a general form of a class of chaotic system, which can be synchronized between a fractional-order chaotic system and an integer order chaotic system. Furthermore, an example is carried out to verify and demonstrate the effectiveness of the proposed control scheme. Simultaneously, our work is supported by logical theorems and intuitive numerical simulation.

Soil salinization has become a highly significant eco-system issue that is encountered all over the world. Serious soil salinization leads to soil deterioration and has a negative impact on sustainable development of the eco-system and agriculture. However, the spectral reflectance of soils with high overlap and indecipherability makes it difficult to classify the soil salinization degree quickly and accurately. In this paper, an innovative, intelligent methodology using a fractional-order chaotic system to classify the soil salinization degree is proposed. To select a suitable order for the fractional-order chaotic system, the integer-order and noninteger order master-slave Lorenz chaotic systems were used to observe variations in the phase plane distributions. Movement traces of the chaotic system show that severely saline soil will exhibit more active changes, and its distribution status of the Lorenz chaotic system will be more scattered. After analyzing the characteristics of phase plane distributions, a preferred 0.9 fractional-order chaotic system is selected to obtain good analytical characteristics. Finally, extenics theory is used to verify the accuracy of salinization status classified by the coordinate values of the chaotic attractors, and an extenic matter element model is established to analyze the salinization degree. From the results, it was found that 100% analysis accuracy in the judgment of salinization level could be achieved under noninteger order status, and this judgment method is also suitable for soils in different human activity areas. This method has now become a benchmark for testing soil salinization with a chaotic system and is an innovative method that can be used to test the soil salinization degree quickly and accurately.

Compared with fractional-order chaotic systems with a large number of dimensions, three-dimensional or integer-order chaotic systems exhibit low complexity. In this paper, two novel four-dimensional, continuous, fractional-order, autonomous, and dissipative chaotic system models with higher complexity are revised. Numerical simulation of the two systems was used to verify that the two new fractional-order chaotic systems exhibit very rich dynamic behavior. Moreover, the synchronization method for fractional-order chaotic systems is also an issue that demands attention. In order to apply the Lyapunov stability theory, it is often necessary to design complicated functions to achieve the synchronization of fractional-order systems. Based on the fractional Mittag–Leffler stability theory, an adaptive, large-scale, and asymptotic synchronization control method is studied in this paper. The proposed scheme realizes the synchronization of two different fractional-order chaotic systems under the conditions of determined parameters and uncertain parameters. The synchronization theory and its proof are given in this paper. Finally, the model simulation results prove that the designed adaptive controller has good reliability, which contributes to the theoretical research into, and practical engineering applications of, chaos.

Time delay frequently appears in many phenomena of real life and the presence of time delay in a chaotic system leads to its complexity. It is of great practical significance to study the synchronization control of fractional-order chaotic systems with time delay. This is because it is closer to the real life and its dynamical behavior is more complex. However, the chaotic system is usually uncertain or unknown, and may also be affected by external disturbances, which cannot make the ideal model accurately describe the actual system. Moreover, in most of existing researches, they are difficult to realize the synchronization control of fractional-order time delay chaotic systems with unknown terms. In this paper, for the synchronization problems of the different structural fractional-order time delay chaotic systems with completely unknown nonlinear uncertain terms and external disturbances, based on Lyapunov stability theory, an adaptive radial basis function (RBF) neural network controller, which is accompanied by integer-order adaptive laws of parameters, is established. The controller combines RBF neural network and adaptive control technology, the RBF neural network is employed to approximate the unknown nonlinear functions, and the adaptive laws are used to adjust corresponding parameters of the controller. The system stability is analyzed by constructing a quadratic Lyapunov function. This method not only avoids the fractional derivative of the quadratic Lyapunov function, but also ensures that the adaptive laws are integer-order. Based on Barbalat lemma, it is proved that the synchronization error tends to zero asymptotically. In the numerical simulation, the uncertain fractional-order Liu chaotic system with time delay is chosen as the driving system, and the uncertain fractional-order Chen chaotic system with time delay is used as the response system. The simulation results show that the controller can realize the synchronization control of the different structural fractional-order chaotic systems with time delay, and has the advantages of fast response speed, good control effect, and strong anti-interference ability. From the perspective of long-term application, the synchronization of different structures has greater research significance and more development prospect than self synchronization. Therefore, the results of this study have great theoretical significance, and have a great application value in the field of secure communication.

The extensive utilization of information and communication technologies nowadays enhances information accessibility and underscores the importance of information and data security. Image encryption is a prevalent technique for safeguarding medical data on public networks, serving a vital function in the healthcare sector. Due to their intricate dynamics, memristors are frequently employed in the creation of innovative chaotic systems that enhance the efficacy of chaos-based encryption techniques. In recent years, chaos-based encryption methods have surfaced as a viable method for safeguarding the confidentiality of transmitted images. Memristor-based Fractional-order chaotic systems (MFOCS) have garnered considerable interest because to their resilience and intricacy. Fractional-order chaotic systems (FOCS) exhibit more intricate dynamics than integer-order chaotic systems. Consequently, the exploration of fractional chaotic systems for the development of picture cryptosystems has gained popularity recently. This research introduces an innovative image encryption framework utilizing a memristor-based fractional chaotic map in conjunction with the Secretary Bird Optimization Algorithm (SBOA) to improve security and resilience against cryptographic threats. The suggested method utilizes the distinctive memory properties and high-dimensional chaotic dynamics of the memristor-based fractional system to produce unpredictable encryption keys. The SBOA is utilized to enhance essential encryption parameters, guaranteeing superior randomness and resilience against statistical and differential assaults. The encryption method comprises a confusion phase, in which pixel positions are randomized using chaotic sequences, succeeded by a diffusion phase, where pixel intensities are altered utilizing optimal key sequences. Performance evaluation is executed by entropy analysis, correlation coefficient tests, NPCR, UACI, and studies of computational complexity. The findings indicate that the suggested method attains elevated entropy, minimal correlation, and robust key sensitivity, rendering it exceptionally resilient against brute-force and differential assaults. Notwithstanding its computing burden from fractional-order chaotic dynamics, the suggested model offers a secure and efficient encryption method appropriate for real-time image protection applications.

In this paper, we focus on the synchronization between integer-order chaotic systems and a class of fractional-order chaotic system using the stability theory of fractional-order systems. A new fuzzy sliding mode method is proposed to accomplish this end for different initial conditions and number of dimensions. Furthermore, three examples are presented to illustrate the effectiveness of the proposed scheme, which are the synchronization between a fractional-order Lu chaotic system and an integer-order Liu chaotic system, the synchronization between a fractional-order hyperchaotic system based on Chen’s system and an integer-order hyperchaotic system based upon the Lorenz system, and the synchronization between a fractional-order hyperchaotic system based on Chen’s system, and an integer-order Liu chaotic system. Finally, numerical results are presented and are in agreement with theoretical analysis.