Fractional-order Hyperchaotic System Research Articles

Image encryption based on a fractional-order hyperchaotic system and fast row-column-level joint permutation and diffusion

Image encryption based on a fractional-order hyperchaotic system and fast row-column-level joint permutation and diffusion

Extreme multistability of fractional-order hyperchaotic system based on dual memristors and its implementation

In this paper, a fractional-order hyperchaotic system based on dual memristors is proposed by introducing flux-controlled and charge-controlled memristors into a simple RLC circuit. Dynamics of the hyperchaotic system are investigated using bifurcation diagrams, Lyapunov exponents spectrum (LEs), phase diagrams, time-domain diagrams, spectral entropy (SE) and C0 complexity. The results show that it has a plane of equilibria and exhibits rich dynamical characteristics, including hyperchaos, homogeneous and heterogeneous extreme multistability. Meanwhile, the transient transition phenomena as well as the effect of parameters on the complexity and chaotic behavior of the system are also studied. Furthermore, the practical implementation of the system is realized through analog and digital circuit. The experimental results validate the correctness of the theoretical analysis and help to make better use of the hyperchaotic system in applications such as secure communications.

A fractional-order hyperchaotic system that is period in integer-order case and its application in a novel high-quality color image encryption algorithm

A new fractional-order 5D hyperchaotic system based on memristor is constructed in this paper, with the speciality that the system exists chaotic and hyperchaotic states in the fractional-order case, while in periodic state in the integer-order. In addition, it has a variety of special phenomena at fractional-order such as infinite initial value range, parameter-dependent offset-boosting and amplitude control, attractor coexistence, and fractional order complexity greater than integer order. The correctness and feasibility of the system is verified by analog circuit simulation and hardware circuit implementation. Combining this system with image encryption algorithms, two new scrambling algorithms and a diffusion algorithm are proposed. And a high-quality encryption scheme that can be applied to a wide range of color images is proposed. The scheme is found to have excellent security after verification by various security analyses and comparison with other literatures. This paper provides a basis for the superiority of fractional-order chaotic systems and provides new methods in the field of image encryption.

An algorithm based on 6D fractional order hyperchaotic system and knight tour algorithm to encrypt image

Abstract The security guarantee of data transmission is becoming more crucial as the frequency of information interchange rises. Ensuring the security of images is essential since they serve as a vital transmission medium. This research suggests an image encryption method that combines the knight tour algorithm with a 6D fractional order hyperchaotic system. First, chaotic sequences are produced using a fractional order hyperchaotic system, which is then utilized to index order and jumble the entire image. To retrieve the image after the second scrambling, choose the knight tour beginning point and run ten rounds of knight tour algorithms on the scrambled image. Thirdly, to maximize the efficiency of picture encryption, employ diffusion methods. The outcomes of the imaging experiment were lastly tested and assessed. The security of the image can be successfully guaranteed by a high-dimensional fractional order hyperchaotic system. This is because its high dimensionality gives it a larger key space than the low dimensional system. This is why it can resist attacks more effectively. After a series of evaluation experiments, it is obvious that this encryption scheme has good encryption performance.

Investigation of the hyperchaos and control in the fractional order financial system with profit margin

This research study proposes a novel hyperchaotic finance system with profit margin and then utilizes the Adomian Decomposition Method (ADM) to tackle the solution of the associated Caputo-derivative based fractional-order hyperchaotic finance system with profit margin. Numerical simulations and analyses, such as the evaluation of Lyapunov exponents (LE), the creation of bifurcation diagrams (BD), the complexity analysis (CA), and the 0-1 test, are used to get a full picture of the system. The results of our study demonstrate the occurrence of periodic, chaotic, and hyperchaotic dynamics in the system. Furthermore, the overall criteria for the existence and the uniqueness of the exact solutions for Caputo fractional-order models are presented. In addition, we present a control methodology for the fractional-order hyperchaotic financial system utilizing the Laplace transform and linear feedback control. Significantly, our research showcases a noteworthy correlation between the analytical results and numerical simulations, emphasizing the soundness and effectiveness of our methodology.

Review, design, stabilization and synchronization of fractional-order energy resources demand-supply hyperchaotic systems using Fractional-order PD-based Feedback Control scheme

Review, design, stabilization and synchronization of fractional-order energy resources demand-supply hyperchaotic systems using Fractional-order PD-based Feedback Control scheme

Design of a single input control signal for fault-tolerant synchronization of fractional order hyperchaotic Lu system

ABSTRACT The authors of this paper used an adaptive-sliding mode controller (A-SMC) to synchronise two fractional-order hyperchaotic (FOHC) Lu systems in the presence of external disturbance and bounded parametric uncertainty with unknown bounds. They employed the Riemann–Liouville fractional order and defined a new fractional-order sliding surface for the FOHC Lu system to determine a proper active control. Additionally, they utilised adaptive laws to estimate uncertainty bounds and unknown disturbance signals. The authors proved the stability of the closed-loop control system using the Lyapunov theory. Simulation results in MATLAB demonstrated the desirable performance of this method in the presence of disturbance and parametric uncertainty.

Correction to: Images encryption based on robust multi-mode finite time synchronization of fractional-order hyper-chaotic Rikitake systems

Correction to: Images encryption based on robust multi-mode finite time synchronization of fractional-order hyper-chaotic Rikitake systems

Images encryption based on robust multi-mode finite time synchronization of fractional-order hyper-chaotic Rikitake systems

Images encryption based on robust multi-mode finite time synchronization of fractional-order hyper-chaotic Rikitake systems

Dual Function Matrix Projective Synchronization for Fractional-order Hyperchaotic Systems

Abstract It is well known that the variability and complexity of projection proportionality factors of Dual Projective Synchronization (DPS) can effectively enhance signal confidentiality. However, in most literatures, the proportionality factors are some simple fixed constants, which can't ensure high security of information. For two pairs of fractional-order hyperchaotic systems (FOHS), how to expand the projection proportionality factors to increase its complexity? Then, our work will propose a new synchronization type, i.e., Dual Function Matrix Projective Synchronization (DFMPS) and realize the DFMPS for FOHS for the first time. Firstly, based on the traditional DPS, we generalize the proportionality factors to a general function matrix depending on time t, present the error functions and define the DFMPS. Then, for FOHS, the active controller and synchronization condition are designed and proved. At the same time, when the system is affected by parameter disturbance, the proposed active controller can eliminate the influence of parameter disturbance to the system's DFMPS, which indicates that the proposed control strategy has strong robustness. Finally, the DFMPS of two pairs of fractional-order hyperchaotic Chen and Rabinovich systems are realized, and synchronizing analysis and system robustness analysis are verified by numerical simulation. Particularly, the DFMPS can be degenerated to dual anti-synchronization, dual complete synchronization, DPS, modified DPS and dual matrix projective synchronization. This work extends the synchronization types for FOHS and offers a useful method to explore DFMPS for other fractional-order systems.

Dynamics and circuit simulation of a fractional-order hyperchaotic system

A hyperchaotic system with fractional terms and fractional-order derivatives is investigated in this paper. Simulations show that different attractors such as equilibrium point, limit cycle and hyperchaotic attractor can be generated by the system. Circuit of fractional-order integrator is designed and it is used to implement the circuit of the studied system. The circuit implementation of the studied system proofs its feasibility. A hyperchaotic system with fractional terms and fractional-order derivatives is investigated in this paper. Simulations show that different attractors such as equilibrium point, limit cycle and hyperchaotic attractor can be generated by the system. Circuit of fractional-order integrator is designed and it is used to implement the circuit of the studied system. The circuit implementation of the studied system proofs its feasibility.

Fixed-Time Sliding Mode Synchronization of Uncertain Fractional-Order Hyperchaotic Systems by Using a Novel Non-Singleton-Interval Type-2 Probabilistic Fuzzy Neural Network

In this study, we proposed a sliding mode control method based on fixed-time sliding mode surface for the synchronization of uncertain fractional-order hyperchaotic systems. In addition, we proposed a novel self-evolving non-singleton-interval type-2 probabilistic fuzzy neural network (SENSIT2PFNN) to estimate the uncertain dynamics of the system. Moreover, an adaptive compensator was designed to eliminate the influences of random uncertainty and fuzzy uncertainty, thereby yielding an asymptotically stable controlled system. Furthermore, an adaptive law was introduced to optimize the consequence parameters of SENSIT2PFNN. The membership layer and rule base of SENSIT2PFNN were optimized using the self-evolving algorithm and whale optimization algorithm, respectively. The simulation results verified the effectiveness of the proposed methods for the synchronization of uncertain fractional-order hyperchaotic systems.

A novel fractional-order hyperchaotic complex system and its synchronization

A novel fractional-order hyperchaotic complex system is proposed by introducing the Caputo fractional-order derivative operator and a constant term to the complex simplified Lorenz system. The proposed system has different numbers of equilibria for different ranges of parameters. The dynamics of the proposed system is investigated by means of phase portraits, Lyapunov exponents, bifurcation diagrams, and basins of attraction. The results show abundant dynamical characteristics. Particularly, the phenomena of extreme multistability as well as hidden attractors are discovered. In addition, the complex generalized projective synchronization is implemented between two fractional-order hyperchaotic complex systems with different fractional orders. Based on the fractional Lyapunov stability theorem, the synchronization controllers are designed, and the theoretical results are verified and demonstrated by numerical simulations. It lays the foundation for practical applications of the proposed system.

Video Transmission Secrecy Improvement Based on Fractional Order Hyper Chaotic System

In the Digital World scenario, the confidentiality of information in video transmission plays an important role. Chaotic systems have been shown to be effective for video signal encryption. To improve video transmission secrecy, compressive encryption method is proposed to accomplish compression and encryption based on fractional order hyper chaotic system that incorporates Compressive Sensing (CS), pixel level, bit level scrambling and nucleotide Sequences operations. The measurement matrix generates by the fractional order hyper chaotic system strengthens the efficiency of the encryption process. To avoid plain text attack, the CS measurement is scrambled to its pixel level, bit level scrambling decreases the similarity between the adjacent measurements and the nucleotide sequence operations are done on the scrambled bits, increasing the encryption. Two stages are comprised in the reconstruction technique, the first stage uses the intra-frame similarity and offers robust preliminary retrieval for each frame, and the second stage iteratively improves the efficiency of reconstruction by integrating inter frame Multi Hypothesis (MH) estimation and weighted residual sparsity modeling. In each iteration, the residual coefficient weights are modified using a mathematical approach based on the MH predictions, and the Split Bregman iteration algorithm is defined to resolve weighted l1 regularization. Experimental findings show that the proposed algorithm provides good compression of video coupled with an efficient encryption method that is resistant to multiple attacks.

Dynamical Analysis and Generalized Synchronization of a Novel Fractional-Order Hyperchaotic System with Hidden Attractor

In this paper, hidden dynamical behaviors in a novel fractional-order hyperchaotic system without an equilibrium point are investigated. It is found that the chaotic system exhibits various hidden behaviors for different parameters, such as the hyperchaotic attractor, the chaotic attractor and the limit cycle. The behaviors are demonstrated via phase portraits and time evolution curves. Moreover, generalized synchronization of the systems is discussed, which can be realized by designing suitable controllers. Numerical simulations are carried out to verify the effectiveness of this synchronization scheme. By analyzing the synchronization performance, it is inferred that the lower the derivative order is, the less time is required to reach synchronization.

Finite-time Synchronization of Fractional-order Energy Resources Demand-Supply Hyperchaotic Systems via Fractional-order Prediction-based Feedback Control Strategy with Bio-inspired Multiobjective Optimization

Abstract The concept of fractional-order control (F-oC) is exploited in this paper to synchronize fractional-order dynamical systems. The addressed systems in this paper reflect the real physical phenomena characterized by the complicated relationship between supply and demand for energy resources in the Shanghai area. Thus, we provide the developed fractional energy resource attractor and the simulation results regarding synchronization under the proposed control law of the same fractional energy resource attractor. Note that most of the synchronization methods achieved excellent performance when dealing with complex continuous systems; however, no method addressed the synchronization problem of fractional-order energy resource systems based on the F-oC and modern optimization techniques, to the best of our knowledge. By designing the finite-time control theory, the finite-time full synchronization of two identical fractional-order energy resources demand-supply hyperchaotic systems (F-oERDSHSs) is investigated due to its performance. The advanced prediction-based fractional-order control law (Pb-FoCL) is established for finite-time synchronization of F-oERDSHSs. The design procedure becomes a multiobjective optimization problem of the knowledge base of the developed controller while satisfying the desired performance requirements. The Finite-Time Stability (F-TS) of the control-loop system is proved by using the finite-time Lyapunov stability theory. Furthermore, the Improved Artificial Hummingbird Algorithm (I-AHA) is used to find an optimal knowledge base of Pb-FoCL while achieving the design constraints. Simulation results are provided to verify the efficiency of the proposed control strategy.

Color medical image cryptography technology based on segmentation and fractional-order hyperchaotic system.

This paper proposes a novel color medical image encryption method based on mean shift algorithm and fractional-order hyperchaotic system. Firstly, color medical images were divided into lesion area, detection area, and edge area by using mean shift clustering. Low-value pixels and zero-value pixels in the edge region are abandoned to improve the encryption efficiency. Secondly, the doctor-patient information is embedded into the pixel matrix of the lesion region by using discrete wavelet transform and singular value decomposition. Thirdly, the pixels of the lesion region and the detection region are processed by using fractional-order hyperchaotic system. Finally, the color medical image embedded with doctor-patient information can be quickly encrypted. Security analysis shows that the encryption method has not only strong robustness and security, but also good performance against various attacks. SPNA and CA results: (a) s1-DE-SPNA, (b) s2-DE-SPNA, (c) s3-DE-SPNA, (d) s4-DE-SPNA; (e) s1-DE-CA, (f) s2-DE-CA, (g) s3-DE-CA, (h) s4-DE-CA.

Rich complex dynamics in new fractional-order hyperchaotic systems using a modified Caputo operator based on the extended Gamma function

In this work, rich complex dynamics are observed in new fractional-order hyperchaotic systems when employing a modified Caputo-operator based on the extended Gamma function. Thus, the new parameter of the extended Gamma function provides higher degree of freedom that plays a vital role in the appearance (or disappearance) of chaotic attractors. Two numerical examples are provided to validate the proposed method. The numerical examples show that various chaotic, hyperchaotic and non-chaotic attractors are obtained when varying the new parameter of the modified Caputo-operator.In conclusion, this work shows that the modified Caputo-operator possesses higher degree of freedom than the classic one. This interesting phenomena result in obtaining rich complex dynamics in the systems that involve this new operator including phase transition from hyperchaotic to chaotic states and also phase transition from chaotic to non-chaotic states. Moreover, we show in the second numerical example that chaotic attractors exist only with the modified Caputo-operator using a specific choice of the system’s parameters, while the system’s counterpart does not show chaotic behaviors when using the classic Caputo-operator. Thus, the fractional systems with the modified Caputo-operator are shown to exhibit stronger chaotic dynamics than their counterparts with the old (classic) Caputo-operator.

The large key space image encryption algorithm based on modulus synchronization between real and complex fractional-order dynamical systems.

This paper constructs and analyzes the dynamical properties of a new fractional-order real hyper-chaotic system and its corresponding complex variable system. A thorough analysis was done by employing stability of equilibrium points, phase plots, Lyapunov spectrum, and bifurcation analysis for the consequences of varying fractional-order derivative and parameter values on the system. For the first time, a modulus synchronization scheme is proposed to synchronize real and complex fractional-order dynamical systems. Based on Lyapunov stability theory, non-linear controllers are designed to achieve the proposed modulus synchronization scheme. A new modulus synchronization encryption algorithm with a large key space size for digital images is introduced for the application. The experimental results and analysis validate the desired algorithm. Also, we compare our result of the new encryption algorithm with the previously published literature and verify the efficacy of the considered scheme. Numerical simulations are given to validate the theoretical analysis

Impulsive Control and Synchronization for Fractional-Order Hyper-Chaotic Financial System

This paper reports a new global Mittag-Leffler synchronization criterion with regard to fractional-order hyper-chaotic financial systems by designing the suitable impulsive control and the state feedback controller. The significance of this impulsive synchronization lies in the fact that the backward economic system can synchronize asymptotically with the advanced economic system under effective impulse macroeconomic management means. Matlab’s LMI toolbox is utilized to deduce the feasible solution in a numerical example, which shows the effectiveness of the proposed methods. It is worth mentioning that the LMI-based criterion usually requires the activation function of the system to be Lipschitz, but the activation function in this paper is fixed and truly nonlinear, which cannot be assumed to be Lipschitz continuous. This is another mathematical difficulty overcome in this paper.