Fractional-order Lorenz System Research Articles

Blockchain with optimal deep learning assisted secure data sharing and classification on future healthcare systems

In the future of healthcare, Blockchain (BC) technology holds immense potential for improving the security and privacy of data. By allowing the secure and immutable storage of medical files and healthcare-related transactions, BC ensured that sensitive medical data remains tamper-proof and open only to authorized parties. Patients have greater control over their data's development, revoking or granting access as required, but healthcare workers can streamline data sharing and ensure the integrity of important data. The decentralized nature of BC networks decreases the risk of centralized data breaches, eventually fostering trust and transparency in the healthcare ecosystems. Conversely, deep learning (DL) has great to revolutionize healthcare diagnostics in the future, offering quick and extremely accurate estimates of medical conditions. This technology has greatly enhanced patient solutions, decreased medical expenses, and improved the burden on medical staff by providing appreciated insights into an extensive range of conditions, from cancer to neurological disorders. With this stimulus, this study presents a novel BC with optimal DL-based secure data sharing and classification (BCODL-SDSC) technique in the future healthcare system. The goal of the BCODL-SDSC technique is to secure and thoroughly examine healthcare data using BC and DL techniques. Primarily, the BCODL-SDSC technique enables BC technology to store and maintain the patient’s data from the procedure of several transactions and enable access control to the various stakeholders. For the security of the medical images, the BCODL-SDSC technique applies the Fractional Order Lorenz system (FOLS) based encryption technique with tuna swarm optimization (TSO) algorithm based optimal key generation process. Finally, a multi-stage process performs the classification of the medical images: MobileNetv1 feature extractor, artificial rabbit’s optimization (ARO) based hyperparameter tuning, and stacked recurrent neural network (SRNN) based classification. The experimental outcome of the BCODL-SDSC technique was examined on a benchmark medical image database. An extensive comparative study reported that the BCODL-SDSC technique reaches an effective performance with other models with a maximum accuracy of 99.11%.

Dynamics of the Fractional-Order Lorenz System Based on Adomian Decomposition Method and Its DSP Implementation

Dynamics and digital circuit implementation of the fractional-order Lorenz system are investigated by employing Adomian decomposition method (ADM). Dynamics of the fractional-order Lorenz system with derivative order and parameter varying is analyzed by means of Lyapunov exponents (LEs), bifurcation diagram, chaos diagram and phase diagram. Results show that the fractional-order Lorenz system has rich dynamical behavior and it is a potential model for application. It is also found that the minimum order is affected by numerical algorithm and time step size. Finally, the fractional-order system is implemented on DSP digital circuit. Phase diagrams generated by the DSP are consistent with that generated by simulation.

Secure blind watermarking using Fractional-Order Lorenz system in the frequency domain

This paper investigates two different blind watermarking systems in the frequency domain with the development of a Pseudo Random Number Generator (PRNG), based on a fractional-order chaotic system, for watermark encryption. The methodology is based on converting the cover image to the YCbCr color domain and applying two different techniques of frequency transforms, Discrete Cosine Transform (DCT) and Discrete Wavelet Transform (DWT), to the Y channel. Then, the encrypted watermark is embedded in the middle-frequency band and HH band coefficients for the DCT and DWT, respectively. For more security and long encryption key size, the fractional-order Lorenz system is used to double the encryption key size and make it secure against brute-force attacks. The proposed algorithms successfully detect the hidden watermark by using the statistical properties of the embedding media, where the PRNG is examined using statistical tests and the watermarking systems are evaluated using standard imperceptibility and robustness measures. Common attacks such as noise-adding attacks, image enhancement attacks and geometric transformation attacks are discussed. Results of the PRNG demonstrate sensitivity to the system parameters, and results of the watermarking systems show good imperceptibility while keeping the robustness measures in a good range.

Dynamical analysis of nonlinear fractional order Lorenz system with a novel design of intelligent solution predictive radial base networks

In this research work, the convective flow of Lorenz attractor in the fractional domain is modeled with a nonlinear flexible structure of radial basis neural network (RBNN). The physical parameters of the complex fractional-order system are initially computed with fractional order Runge–Kutta solver for different stochastic scenarios of control constraints and designed its parametric model with RBFN under various initial conditions of the chaotic Lorenz system. Multiple chaos of the Lorenz system with the dynamical neural network structure are computed for different fractional order solutions to analyze the sensitivity of chaotic behavior by means of Lyapunov exponents. Phase diagrams of the chaotic pattern show the fractional-order Lorenz system has compact dynamical behavior and has the potential for the design application of a real dynamical system. Embedded dimension and time delay of the fractional Lorenz system are computed using the average mutual information (AMI) technique to evaluate the suitability of the time delay chaotic pattern in the fractional domain, which is affected by the numerical algorithm and fluctuated on minute time step size. The proposed design of RBFN has been confirmed as an exceptionally high-performance tool for soft computing and dynamics analysis of chaotic systems in the fractional order domain.

Data-driven discovery of Caputo fractional order systems

Due to the peculiar non-locality of fractional order differential and integral operators, Caputo fractional order systems are harder than integer order systems to be discoverd from data. To solve this open problem, we propose a framework of method capable of discovering Caputo fractional order (autonomous and non-autonomous) systems from measurable data. The interior point method and genetic algorithm are embedded respectively in the framework. The former is mainly presented in text, while the latter is implemented for comparison and validation. The framework is designed to dynamically and coordinately update the fractional order and vector field function for the system to be discovered till the difference between the measured and discovered systems is minimized. It is computationally efficient, robust and illustrated by discovering the Caputo fractional order Lorenz system, Chua’s circuit and Duffing’s oscillator hidden in measured data. As thus, this work provides one way to uncover underlying Caputo fractional order mathematical models (or physical laws and governing equations).

Blockchain-Driven Image Encryption Process with Arithmetic Optimization Algorithm for Security in Emerging Virtual Environments

The real world is bounded by people, hospitals, industries, buildings, businesses, vehicles, cognitive cities, and billions of devices that offer various services and interact with the world. Recent technologies, including AR, VR, XR, and the digital twin concept, provide advanced solutions to create a new virtual world. Due to the ongoing development of information communication technologies and broadcast channels, data security has become a major concern. Blockchain (BC) technology is an open, decentralized, and transparent distributed database that can be maintained by the group. BC’s major features are high credibility, decentralization, transparency, versatility, autonomy, traceability, anonymity, intelligence, reward mechanisms, and irreversibility. This study presents a blockchain-driven image encryption technique using arithmetic optimization with a fractional-order Lorenz system (BDIE-AOFOLS). The BDIE-AOFOLS technique uses the FOLS method, which integrates the Arnold map, tent map, and fractional Lorenz system. Besides this, an arithmetic optimization algorithm (AOA) was carried out for the optimum key generation process to achieve the maximum PSNR value. The design of an AOA-based optimal generation of keys for the FOLS technique determines the novelty of the current work. Moreover, the cryptographical pixel values of the images can be stored securely in the BC, guaranteeing image security. We compared the outcomes of the proposed BDIE-AOFOLS technique against benchmark color images. The comparative analysis demonstrated the improved security efficiency of the BDIE-AOFOLS technique over other approaches, with a mean square error of 0.0430 and a peak signal-to-noise ratio of 61.80 dB.

Determining Lyapunov exponents of fractional-order systems: A general method based on memory principle

Lyapunov exponents provide quantitative evidence for determining the stability and classifying the limit set of dynamical systems. There are several well-established techniques to compute Lyapunov exponent of integer-order systems, however, these techniques failed to generalize to fractional-order systems due to the nonlocality of fractional-order derivatives. In this paper, a method for determining the Lyapunov exponent spectrum of fractional-order systems is proposed. The proposed method is rigorously derived based on the memory principle of Grünwald–Letnikov derivative so that it is generally applicable and even well compatible with integer-order systems. Three classical examples, which are the fractional-order Lorenz system, fractional-order Duffing oscillator, and 4-dimensional fractional-order Chen system, are respectively employed to demonstrate the effectiveness of the proposed method for incommensurate, nonautonomous and low effective order systems as well as hyperchaotic systems. The simulation results suggest that the proposed method is indeed superior to the existing methods in accuracy and correctness.

Solution of a new high-performance fractional-order Lorenz system and its dynamics analysis

Solution of a new high-performance fractional-order Lorenz system and its dynamics analysis

Integrated Circuit Design of Fractional-Order Chaotic Systems Optimized by Metaheuristics

Nowadays, a huge amount of research is done on introducing and implementing new fractional-order chaotic systems. In the majority of cases, the implementation is done using embedded hardware, and very seldom does it use integrated circuit (IC) technology. This is due to the lack of design automation tools ranging from the system level down to layout design. At the system level, the challenge is guaranteeing chaotic behavior by varying all parameters while optimizing dynamical characteristics, such as the Lyapunov spectrum and the Kaplan–Yorke dimension. Using embedded hardware, the implementation is straightforward, but one must perform a scaling process for IC design, in which the biases may be lower than 1 volt but the amplitudes of the state variables of the chaotic systems can have values higher than one. In this manner, this paper describes three levels of abstraction to design fractional-order chaotic systems: The first one shows the optimization of a case study, the mathematical model of the fractional-order Lorenz system to find the fractional-orders of the derivatives, and the coefficients that generate better chaotic behavior. The second level is the block description of a solution of the mathematical model, in which the fractional-order derivatives are approximated in the Laplace domain by several approximation methods. The third level shows the IC design using complementary metal–oxide–semiconductor (CMOS) technology. The transfer functions approximating the fractional-order derivatives are synthesized by active filters that are designed using operational transconductance amplifiers (OTAs). The OTAs are also used to design adders and subtractors, and the multiplication of variables is done by designing a CMOS four-quadrant multiplier. The paper shows that the simulation results scaling the mathematical model to have amplitudes lower than ±1 are in good agreement with the results using CMOS IC technology of 180 nm.

On the Stability of Commensurate Fractional-Order Lorenz System

On the Stability of Commensurate Fractional-Order Lorenz System

Hyperbolic uncertainty estimator based fractional order sliding mode control framework for uncertain fractional order chaos stabilization and synchronization

This paper formulates a new fractional order (FO) integral terminal sliding mode control algorithms for the stabilization and synchronization of N-dimensional FO chaotic/hyper-chaotic systems, which are perturbed with unknown uncertainties. In order to render closed loop robustness, a novel efficient double hyperbolic functions based uncertainty estimator is developed for the estimation and mitigation of unknown uncertainties. Moreover, a double hyperbolic reaching law comprising of tangent hyperbolic and inverse sine hyperbolic functions is incorporated in the presented control techniques for the practical convergence of various chaotic system states and tracking errors to infinitesimally close to equilibrium. Examples such as FO Lu, FO Chen and FO Lorenz systems are taken to investigate robustness, finite time convergence, tracking accuracy and closed loop stability properties of the devised methodologies. Last but not least, comparative analysis is also carried out between the proposed and prior control techniques through various time domain performances such as settling time, error indices and measure of control energy.

Synchronization of Fractional-Order Chaotic Systems with Model Uncertainty and External Disturbance

This paper is concerned with complete synchronization of fractional-order chaotic systems with both model uncertainty and external disturbance. Firstly, we propose a new dynamic feedback control method for complete synchronization of fractional-order nominal systems (without both uncertainty and disturbance). Then, a new uncertainty and disturbance estimator (UDE)-based dynamic feedback control method for the fractional-order systems with both uncertainty and disturbance is presented, by which the synchronization problem of such fractional-order chaotic systems is realized. Finally, the fractional-order Lorenz system is used to demonstrate the practicability of the proposed results.

A novel colour image encryption based on fractional order Lorenz system

A colour image encryption algorithm based on fractional Lorenz system has been developed in this paper. Firstly, the colour image has been divided into three layers: R, G, and B, and Arnold map has been used for scrambling the three layers separately, and the plaintext pixel has been selected as the scrambling parameter. The Tent map has been used to associate the plain text to generate the initial values of the fractional Lorenz system, which has been iterated by the system to generate a pseudo-random sequence that can be used for two pixel diffusions. The first pixel diffusion is to diffuse the pixels of each layer to the other two layers, and the second pixel diffusion is to add modulus diffusion inside the layer. Simulation results illustrate that the plaintext image is encrypted into a meaningless noise image. The superior performance of proposed algorithm has been proved by the key space analysis, histogram analysis, correlation test, sensitivity analysis and information entropy analysis, and it can resist various common attack methods.

Dynamic characteristics analysis of time-delay fractional order dynamic system

In this paper, a kind of time-delay fractional order dynamic system is studied. The classical integral sliding mode control method is used to design a single controller to control the time-delay fractional order Liu system from chaos to fixed point, and the single controller is further designed to control the time-delay fractional order Liu system from chaotic attractor to limit loop, so the chaos control of time-delay fractional order nonlinear dynamic system is achieved. On that basis, the dynamic characteristics of the time-varying delayed fractional-order Lorenz system are analyzed by the simulation, and the chaotic phenomena of the time-varying delayed fractional-order Lorenz system are verified under the provided time delay and order.

Stability Analysis and Synchronization of Fractional Order Complex Lorenz System

Chaotic nature of the fractional-order system and synchronization have been two very significant problems in the nonlinear sciences, even in the complex fractional-order system. In the present manuscript, a study of the fractional-order complex Lorenz system which exhibits a chaotic behavior would be analyzed by numerical simulation results. Stability analysis of fractional-order system has been studied and obtains some useful results for the researcher working in the nonlinear dynamics. Synchronization of the proposed systems is also studied and it has been achieved by active control method.

Non-fragile exponential polynomial observer design for a class of nonlinear fractional-order systems with application in chaotic communication and synchronisation

ABSTRACTThis paper addresses the design of a non-fragile exponential polynomial observer for a class of fractional-order nonlinear systems. Existence of the observer is proven and a sufficient condition for the stability of the state estimation error dynamics with a predetermined exponential convergence rate is derived employing the Lyapunov stability theorem. The exponential stability criterion is proposed in the form of linear matrix inequalities (LMIs). Moreover, some numerical examples have been provided to illustrate the effectiveness of the proposed approach. The synchronisation of fractional-order Lorenz systems has been investigated using the proposed method. Then, the proposed method has been applied to the chaotic communication problem of fractional-order chaotic systems with four scroll attractors.

Parameter estimation for fractional-order chaotic systems by improved bird swarm optimization algorithm

The essence of parameter estimation for fractional-order chaotic systems is a multi-dimensional parameter optimization problem, which is of great significance for implementing fractional-order chaos control and synchronization. Aiming at the parameter estimation problem of fractional-order chaotic systems, an improved algorithm based on bird swarm algorithm is proposed. The proposed algorithm further studies the social behavior of the original bird swarm algorithm and optimizes the foraging behavior in the original bird swarm algorithm. This method is applied to parameter estimation of fractional-order chaotic systems. Fractional-order unified chaotic system and fractional-order Lorenz system are selected as two examples for parameter estimation systems. Numerical simulation shows that the algorithm has better convergence accuracy, convergence speed and universality than bird swarm algorithm, artificial bee colony algorithm, particle swarm optimization and genetic algorithm.

Anti-Synchronization of a Class of Chaotic Systems with Application to Lorenz System: A Unified Analysis of the Integer Order and Fractional Order

The paper proves a unified analysis for finite-time anti-synchronization of a class of integer-order and fractional-order chaotic systems. We establish an effective controller to ensure that the chaotic system with unknown parameters achieves anti-synchronization in finite time under our controller. Then, we apply our results to the integer-order and fractional-order Lorenz system, respectively. Finally, numerical simulations are presented to show the feasibility of the proposed control scheme. At the same time, through the numerical simulation results, it is show that for the Lorenz chaotic system, when the order is greater, the more quickly is anti-synchronization achieved.

Detecting chaos in fractional-order nonlinear systems using the smaller alignment index

The identification between chaos and ordered states in fractional-order chaotic systems is a challenge as well as a hot topic due to the complex of fractional calculus. In this paper, the smaller alignment index (SALI) is developed to detect chaos in the fractional-order chaotic systems by introducing the fractional-order tangent systems. Numerical simulations are carried out based on the fractional-order simplified Lorenz system and the fractional-order Hénon map, which are continuous chaotic system and discrete chaotic system, respectively. It shows that the proposed method is effective for distinguishing chaos and order in different kinds of fractional-order chaotic systems.

What is the lowest order of the fractional-order chaotic systems to behave chaotically?

Considering the fractional-order simplified Lorenz system as an example, we analyze the lowest order under the effects of variations in the numerical methods employed to solve the system. We varied the commensurability of the system equations, the system parameter and iteration step to reveal how they affect the appearance of chaos. The results show that the lowest order is obtained using the Adomian decomposition method (ADM), and it is smaller than that obtained through the Adams-Bashforth-Moulton (ABM) algorithm. The fractional-order system in the case of incommensurate order always has a smaller lowest order for chaos than does the system with commensurate order. We found that the lowest order of the fractional-order system decreases as the system parameter increases or as the iteration step decreases. In addition, by selecting the optimal results from a previous analysis, we obtain the lowest order of the fractional-order simplified Lorenz system is 1.136, which is lower than all results previously reported. Finally, we propose a corollary: the lowest order of a fractional-order chaotic system returned by an exact solution condition is lower than the approximate solution derived by numerical simulations. This work facilitates us to find a lower lowest order of fractional-order nonlinear systems, which are relevant to the study of fractional-order nonlinear system for engineering application.