Fractional Hyperchaotic Systems Research Articles

Comparative Analysis of the Chaotic Behavior of a Five-Dimensional Fractional Hyperchaotic System with Constant and Variable Order

Comparative Analysis of the Chaotic Behavior of a Five-Dimensional Fractional Hyperchaotic System with Constant and Variable Order

Experimental validation of IoT image encryption scheme based on a 5-D fractional hyperchaotic system and Numba JIT compiler

In the realm of Internet of Things (IoT) systems, the interconnectivity of physical hardware devices is a fundamental aspect, and as a result, data exchange assumes a critical role in the network. Given the sensitivity of such information, it is imperative to adopt appropriate measures to encrypt the data to safeguard it from unauthorized access. It is, therefore, paramount to prove novel encryption algorithms at an experimental level. To overcome that, a potential solution is fractional-order multi-scroll chaotic systems, which provide an extra degree of freedom to enhance ergodicity and random-like behaviors, which can help to improve encryption keyspace to get robust ciphers. However, the numerical algorithms to obtain a fractional chaotic series increase the computational cost because they demand extensive simulation time to model the memory of the standard fractional derivatives, limiting high-speed encryption. In this framework, a fast encryption scheme using a 5D fractional-order (FO) hyper-chaotic multi-scroll (HCMS) system is proposed and verified experimentally. Based on multiprocessing strategies and the Numba just-in-time (JIT) compiler, the Python code that describes the FO-HCMS system is optimized, which enables image encryption in real-time. The physical implementation of the encryption scheme is performed on an Advance RISC Machine (ARM) processor, and it is applied for image encryption on a machine-to-machine (M2M) communication using the message queuing telemetry transport (MQTT) protocol. The obtained results indicate that the proposed encryption scheme can reach encryption throughputs of up to 19.891 Mbps on an ARM with a 1.4 GHz CPU and 77.864 Mbps on a PC with a 3.1 GHz CPU.

Fractional hyper-chaotic system with complex dynamics and high sensitivity: Applications in engineering

Hyper-chaotic systems have useful applications in engineering applications due to their complex dynamics and high sensitivity. This research is supposed to introduce and analyze a new noninteger hyper-chaotic system. To design its fractional model, we consider the Caputo fractional operator. To obtain the approximate solutions of the extracted system under the considered fractional-order derivative, we employ an accurate nonstandard finite difference (NSFD) algorithm. Moreover, the existence and uniqueness of the solutions are provided using the theory of fixed-point. Also, to see the performance of the utilized numerical scheme, we choose different values of fractional orders along with various amounts of the initial conditions (ICs). Graphs of solutions for each case are provided.

Nonlinear analysis of a four-dimensional fractional hyper-chaotic system based on general Riemann-Liouville-Caputo fractal-fractional derivative.

In this study, a four-dimensional fractional hyperchaotic model is analyzed based on general Riemann–Liouville–Caputo (RLC) fractal–fractional derivative (FFD). A series of new operators are constructed using three different elements, namely, the general Mittag–Leffler function, exponential decay, and power law. The operators have two parameters: One is considered as fractional order and the other as fractal dimension. The Qi hyperchaotic fractional attractor is modeled by using these operators, and the models are solved numerically using a very efficient numerical scheme. Meanwhile, the existence and uniqueness of solutions have been investigated to justify the physical adequacy of the model and the numerical scheme proposed in the resolution. The numerical simulations for some specific fractional order and fractal dimension are presented. Furthermore, these results obtained via generalized Caputo–Fabrizio and fractal–fractional derivative show some crossover effects, which is due to non-index law property. Finally, these obtained from generalized fractal–fractional derivative show very strange and new attractors with self-similarities.

A new image encryption scheme based on fractional-order hyperchaotic system and multiple image fusion

A multi-image encryption scheme based on the fractional-order hyperchaotic system is designed in this paper. The chaotic characteristics of this system are analyzed by the phase diagram, Lyapunov exponent and bifurcation diagram. According to the analyses results, an interesting image encryption algorithm is proposed. Multiple grayscale images are fused into a color image using different channels. Then, the color image is scrambled and diffused in order to obtain a more secure cipher image. The pixel confusion operation and diffusion operation are assisted by fractional hyperchaotic system. Experimental simulation and test results indicate that the devised multi-image encryption scheme can effectively encrypt multiple images, which increase the efficiency of image encryption and transmission, and have good security performance.

An Asymmetric Image Encryption Algorithm Based on a Fractional-Order Chaotic System and the RSA Public-Key Cryptosystem

Herein, an asymmetric image encryption algorithm based on RSA cryptosystem and a fractional-order chaotic system is proposed. Its security depends on RSA algorithm. First, a pair of public and private keys is generated by RSA algorithm. Subsequently, a random message shown as plaintext key information is encrypted by the public key and RSA to achieve ciphertext key information. Next, a new transformation map is established to generate the initial key according to the ciphertext key information. Subsequently, the initial key is substituted into a fractional hyperchaotic system equation to calculate the keystream. Finally, permutation and diffusion operations are employed to encrypt a plain image to obtain the final cipher image. In the proposed algorithm, different keys for encryption and decryption are designed under an asymmetric architecture. The RSA algorithm and fractional chaotic system are combined to encrypt images; in particular, a fast algorithm for computing power multiplication is employed, which significantly improves the encryption effect and enhances the security. Simulation results show that the proposed algorithm is effective and applicable to image protection.

Double image compression-encryption algorithm based on fractional order hyper chaotic system and DNA approach

In order to improve the efficiency and security of image compression-encryption algorithms, we propose a double image compression-encryption scheme based on fractional hyper-chaotic system and DNA approach. Firstly, two images are processed by discrete cosine transform. Secondly, the spectrums of the two images are sorted by Z-scan, so that the two images can be compressed and mixed into a new image. Finally, the resulting image is encrypted by using DNA coding. Different from traditional image encryption algorithms, the proposed algorithm provides a variety of DNA coding and operation modes. Chaotic sequences are used to control the coding and operation mode in order to improve the complexity of the encryption process. Fractional order and initial values of fractional order hyper-chaotic system are used as the key of the proposed algorithm, which greatly expands the complexity and the key space of the scheme. In the proposed scheme, by mixing two images and performing a compression-encryption operation on them simultaneously, the proposed algorithm can improve the complexity of encrypted images while providing good confusion. Experimental results and security analysis show that the proposed algorithm can effectively resist multiple attacks.

Λ − φ generalized synchronization: application to fractional hyperchaotic systems with arbitrary dimensions and orders

This paper investigates the generalized synchronization between non-identical fractional-order systems characterized by different dimensions and different orders. The generalized synchronization combines the inverse matrix projective synchronization with the generalized synchronization. In particular, the proposed approach enables generalized synchronization to be achieved between n-dimensional master system and m-dimensional slave system in different dimensions. The technique, which exploits nonlinear controllers, stability property of integer-order linear systems and Lyapunov stability theory, proves to be effective in achieving the generalized synchronization. Finally, the approach is applied between 4-D and 5-D fractional hyperchaotic systems with the aim to illustrate the capabilities of the novel scheme proposed herein.

Anticontrol of a Fractional-Order Chaotic System and Its Application in Color Image Encryption

This paper investigates the anticontrol of the fractional-order chaotic system. The necessary condition of the anticontrol of the fractional-order chaotic system is proposed, and based on this necessary condition, a 3D fractional-order chaotic system is driven to two new 4D fractional-order hyperchaotic systems, respectively, without changing the parameters and fractional order. Hyperchaotic properties of these new fractional dynamic systems are confirmed by Lyapunov exponents and bifurcation diagrams. Furthermore, a color image encryption algorithm is designed based on these fractional hyperchaotic systems. The effectiveness of their application in image encryption is verified.

Image compression-encryption scheme based on fractional order hyper-chaotic systems combined with 2D compressed sensing and DNA encoding

An image compression-encryption scheme is proposed based on fractional order hyper-chaotic systems combined with 2D compressed sensing (CS) and DNA encoding. This scheme adopts the CS technique to cut down the size of the original image proportionally in the encryption process and the construction parameters of the CS measurement matrix are controlled by fractional order hyper-chaotic systems. The chaotic sequences generated by fractional order hyper-chaotic systems are used to dynamically control DNA encoding and DNA operation mode to improve the complexity of encryption process. The fractional order and initial value of the fractional hyper-chaotic system are used as the key of the encryption scheme, which greatly expands the key space of the proposed scheme. The proposed scheme maximizes the utilization of chaotic sequences generated by fractional order hyper-chaotic systems, and the chaotic sequences fit perfectly with the whole process of the scheme. The experimental results and security analysis show the security of the scheme against multiple attacks.

Research on realization and optimization of communication equipment security engineering circuit for nonlinear system based on fractional differential operator.

Compared with integral calculus, the fractional differential operator can objectively reveal and describe the physical characteristics of the actual system. For fractional differential operator functions, sufficient conditions for stability of fractional nonlinear systems are given. By accurately adjusting the frequency of the analog input signal and observing and verifying the nonlinear dynamic characteristics of the new system, the simulation experiment of the fractional circuit with different fractional values is carried out, and the circuit simulation can visually observe the evolution of system variables. The research shows that the predictive correction method numerically simulates the fractional-order system, and the phase diagram of the chaotic attractor of the system is obtained. The simulation results show that the minimum order of chaos in the fractional hyperchaotic system is 2.8. The research shows that the simulation of the nonlinear system and its circuit implementation show the effectiveness of the circuit simulation method of the fractional-order chaotic system and the feasibility of circuit implementation.

Control of a novel fractional hyperchaotic system using a located control method

Fractional order dynamics and chaotics systems have been recently combined, yielding interesting behaviours. In this paper, a novel integer order hyperchaotic system is considered. Then, a fractional order hyperchaotic representation of said system is proposed using a natural fractionalization. Two different linear control methodologies to deal with the complexity which introduce such systems are proposed. Those methods are able to modify the hyperchaotic behaviour of the system and force it to move towards a fixed point; i.e. steady state. These approaches give a general framework for taming such complex systems using simple linear controllers. The main tools for analysing the controlled system are Matignon stability criterion and RouthHurwitz test. Using a reliable numerical simulation, the designed system is simulated to verify the theoretical analysis.

Synchronization of Fractional-Order Hyperchaotic Systems via Fractional-Order Controllers

In this paper, the synchronization of fractional-order chaotic systems is studied and a new fractional-order controller for hyperchaos synchronization is presented based on the Lyapunov stability theory. The proposed synchronized method can be applied to an arbitrary four-dimensional fractional hyperchaotic system. And we give the optimal value of control parameters to achieve synchronization of fractional hyperchaotic system. This approach is universal, simple, and theoretically rigorous. Numerical simulations of several fractional-order hyperchaotic systems demonstrate the universality and the effectiveness of the proposed method.

Adaptive Synchronization of Fractional Hyper-Chaotic System with Unknown Parameters

In this paper, chaos synchronization of the modified Sprott E system is investigated. Based on the stability theorem for fractional systems, tracking control approach is used for the fractional-order systems with uncertain parameters. Meanwhile, suitable adaptive synchronization controller and recognizing rules of the uncertain parameters are designed. Numerical simulation results show that the method is easy to implement and reliable for synchronizing the two nonlinear fractional order hyper-chaotic systems.

Full state hybrid projective synchronization of a novel incommensurate fractional order hyperchaotic system using adaptive mechanism

In this paper, full state hybrid projective synchronization of two new incommensurate fractional hyperchaotic systems are presented. The synchronization is achieved under a master–slave configuration in which these systems have different fractional orders. The synchronization scheme and control technique are performed subject to parameter uncertainty in both master and slave systems. The main idea of such asymptotical synchronization is an adaptive mechanism which employs Lyapunov stability criterion. Numerical simulations support the proposed techniques.

Projective Synchronzing a Novel Fractional Hyperchaotic System Based on a Approach Configuring a Special Matrix

This work presents a new approach configuring a special matrix to design controller for synchronizing fractional chaotic systems. With this method, fractional chaotic projective synchronization is implemented. Numerical simulations confirm the effectiveness of the approach.

Fractional sliding mode surface controller for projective synchronization of fractional hyperchaotic systems

In this paper, a sliding mode controller with fractional sliding surface is designed for projective synchronization (PS) of fractional hyperchaotic systems via active control theory and sliding mode theory. The existence and the stability of PS between two fractional hyperchaotic systems are studied based on Lyapunov theory, fractional stability theory and the theory of fractional calculus. The criterions for the practical stability of the PS error system are presented also. Numerical simulation of PS between fractional Chen system and a new hyperchaotic system shows the effectiveness of the proposed controller law.

Adaptive synchronization between different fractional hyperchaotic systems with uncertain parameters

This paper addresses parameter identification and the adaptive synchronization between fractional CYQY hyperchaotic system and fractional Lorenz hyperchaotic system with uncertain parameters. Based on fractional stability theory，an adaptive controller and the paramter updating rule are designed. Numerical simulation shows the effectiveness of the adaptive controller.