Fractional-order Systems Research Articles

Hidden complex multistable dynamical analysis and FPGA implementation of integer-fractional order memristive-memcapacitive chaotic system

Abstract A chaotic circuit based on a magnetic-controlled memristor and charge-controlled memcapacitor is proposed in this paper. The study reveals that it is a hyperchaotic system with hidden characteristics in integer-order. Furthermore, as the parameters change, the attractors exhibit rich evolutionary phenomena. Even after adjusting some parameters to very large values, the system still maintains hyperchaotic behavior. Interestingly, the basin of attraction shows the multistability of the system. Under initial value control, coexisting attractors are categorized into two types: those with initial offset-boosting behavior and nested attractors. When under parameter control, coexisting attractors are divided into two types: symmetric coexisting attractors and nested coexisting attractors. By analyzing the spectral entropy (SE) complexity of the system and using a complexity distribution diagram with two parameters and two initial values, the existence of multiple complex dynamic behaviors in the system has been verified. The fractional-order memristive-memcapacitive system based on the Grunwald-Letnikov algorithm and the five fractional-order values of q i (i = 1, 2, 3, 4, 5) are taken as different in the numerical simulation, it also displays multiple coexisting phenomena similar to those of the integer-order. Finally, Matlab/Simulink and DSP Builder software platform are used to design the fractional-order five-dimensional chaotic memristive-memcapacitive system, and then combined with VHDL and Verilog HDL hardware language, the proposed circuit system is verified on the EP4CE115F29C7 FPGA main chip of Cyclone IV E series. The consistency of hardware implementation and software simulation shows the correctness and feasibility of the design.

Analysis of a fractional order epidemiological model for tuberculosis transmission with vaccination and reinfection.

This study has been carried out using a novel mathematical model on the dynamics of tuberculosis (TB) transmission considering vaccination, endogenous re-activation of the dormant infection, and exogenous re-infection. We can comprehend the behavior of TB under the influence of vaccination from this article. We compute the basic reproduction number ( ) as well as the vaccination reproduction number ( ) using the next-generation matrix (NGM) approach. The theoretical analysis demonstrates that the disease-free equilibrium point is locally asymptotically stable, and the fractional order system is Ulam-Hyers type stable. We perform numerical simulation of our model using the Adams-Bashforth 3-step method to verify the theoretical results and to show the model outputs graphically. By performing data fitting, we observe that our formulated model produces results that closely match real-world data. Our findings indicate that vaccinating a limited segment of the population can effectively eradicate the disease. The numerical simulations also show that vaccination can reduce the number of susceptible and infectious individuals in the population. Moreover, the graphical representations illustrate that the number of infected individuals rises due to both exogenous reinfection and endogenous reactivation.

Observer-Based Prescribed Performance Adaptive Neural Network Tracking Control for Fractional-Order Nonlinear Multiple-Input Multiple-Output Systems Under Asymmetric Full-State Constraints

In this work, the practical prescribed performance tracking issue for a class of fractional-order nonlinear multiple-input multiple-output (MIMO) systems with asymmetric full-state constraints and unmeasurable system states is investigated. A neural network (NN) nonlinear state observer is developed to estimate the unmeasurable states. Furthermore, the barrier Lyapunov functions with the settling time regulator are employed to deal with the asymmetric full-state constraint from the fractional-order MIMO system. On this ground, the prescribed performance adaptive tracking control approach is designed, assuring that all system states do not exceed the prescribed boundaries, and the tracking errors converge to the predetermined compact sets within a predefined time. Finally, two simulation examples are presented to show the effectiveness and practicability of the proposed control scheme.

Intelligent-based approach for parameters identification and control of fractional order systems

The optimization algorithms for identification and control processes take more and more place in the industrial domain. However, determining a control law for fractional systems remains challenging for researchers today. This paper presents a simple method for using a fractional PID (FOPID) corrector to regulate a class of fractional systems that show a stable step response. The procedure is based on an algorithmic identification to assimilate the fractional system into a simple model. Optimization methods were used. Using techniques like Particle Swarm Optimization (PSO), Genetic Algorithm (GA), Artificial Bee Colony (ABC), and Ant Colony Optimization (ACO) to ascertain the parameters of the basic model, a first-order system with delay. Then, the FOPID controller will be calculated using Ziegler–Nichols methods and optimization algorithms to stabilize the real process. Better performance Overshoot and Settling time as well as minimal Energy control effort are benefits of the proposed design. To confirm that the fractional regulator is robust which is optimized by the best-chosen method (PSO), the parameters of the chosen process will be randomly varied. Through two numerical simulations, the efficiency of the suggested method has been confirmed.

Controllability of Impulsive Damped Fractional Order Systems Involving State Dependent Delay

In this article, the concept of controllability on fractional order impulsive systems involving state dependent delay and damping behavior is analysed by utilizing Caputo fractional derivative. The main motivation is to derive the sufficient conditions for the controllability of the considered systems. Based on the Laplace transform and inverse Laplace transform, the solution of fractional-order dynamical systems are obtained. The results are established by utilizing basic ideas of fractional calculus, Mittag-Leffler function and Banach fixed point theorem. Finally, an application is provided to illustrate the derived result.

Stability Analysis Techniques for Commensurate Fractional-Order Systems with Caputo Derivatives

Stability Analysis Techniques for Commensurate Fractional-Order Systems with Caputo Derivatives

Modified Mikhailov stability criterion for non-commensurate fractional-order neutral differential systems with delays

This paper studies the asymptotic stability of non-commensurate fractional-order neutral differential systems with constant delays. To do this, we propose a modified Mikhailov stability criterion. Our work not only generalizes the existing results in the literature but also provides a rigorous mathematical basis for the frequency domain analysis method concerning fractional-order systems with delays. Specific examples and numerical illustrations are also provided to demonstrate the validity of the obtained result.

Securing Consensus in Fractional-Order Multi-Agent Systems: Algebraic Approaches Against Byzantine Attacks

This paper investigates the behavior of fractional-order nonlinear multi-agent systems subjected to Byzantine assaults, specifically focusing on the manipulations of both sensors and actuators. We employ weighted graphs, both directed and undirected, to illustrate the system's topology. Our methodology combines algebraic graph theory with fractional-order Lyapunov techniques to develop algebraic requirements for leader-following consensus, providing a robust framework for analyzing consensus dynamics in these complex systems. We present quantitative results demonstrating the effectiveness of our approach, including two numerical examples that validate the proposed requirements for consensus evaluation. Notably, our work highlights the novelty of using fractional-order systems to enhance resilience against adversarial conditions, contributing significantly to the field of multi-agent systems. By clarifying key terms and streamlining our language, we ensure accessibility for a broader audience while emphasizing the implications of our findings for real-world applications.

Hopf bifurcation control of memristor-based fractional delayed tri-diagonal bidirectional associative memory neural networks under various controllers

In this paper, authors present a memristor-based fractional order system of tri-diagonal bidirectional associative memory neural networks (TdBAMNNs) incorporating leakage and communication delays. The existence and uniqueness theorem is established for the memristor-based fractional-order TdBAMNNs system. Analysis of Hopf bifurcation anti-control is conducted using various feedback controllers by exploring single-parameter and double-parameter approaches. A study of the stability switching curves is undertaken to determine the direction of stability transitions in the system. The theoretical results and graphical illustrations are validated through numerical simulations to exhibit their feasibility. Comparative analysis of the system’s dynamical behavior under different controllers is performed. This paper comprehensively explores the proposed memristor-based system, proving its theoretical underpinnings, and analyzing stability through bifurcation control strategies and comparative assessments under different control scenarios.

Multi-step Residual Power Series Method for Solving Stiff Systems

In this paper, an efficient algorithm based on the residual power series method (RPSM) is presented to solve stiff systems of Caputo fractional order. We apply the RPSM on subintervals to get approximate solutions of these types of systems. The RPSM has advantages that it is suitable to solve linear and nonlinear systems and it gives high accurate results. Modifying this technique to multi-step RPSM considerably reduces the number of arithmetic operations and so reduces the time, especially when dealing with Stiff systems. Several numerical examples are given to show the efficiency, simplicity and the accuracy of the proposed method. Comparing classical RPSM with the new multi-step scheme shows that multi-step RPSM controls the convergence behaviour of the stiff systems. That is, the comparison reveals that MS-FRPSM reduces both absolute and residual errors. More iterations and a smaller step size lead to higher accuracy. Moreover, in MS-FRPSM, the intervals of convergence for the series solution will increase.

The Impact of White Noise on Chaotic Behavior in a Financial Fractional System with Constant and VariableOrder: A Comparative Study

In this study, we investigated the impact of white noise on the chaotic dynamics of a financial system with Caputo fractional derivatives of constant- and variable-order. We solve thesefractional financial systems numerically. We analyze the chaotic behavior of these fractional-order hyperchaotic systems through simulations, study the effects of white noise on chaotic dynamics, and provide a comparison between fractional hyperchaotic systems of constant and variable orders. The study reveals that white noise can either amplify or suppress chaos, with significant differences observed between the constant- and variable-order cases. We obtained numerous intriguing graphical findings for the model by evaluating various scenarios. The findings provide useful information for better understanding financial market dynamics in the presence of noise.

Cauchy discretization method for solve fractional linear optimal control problems

This paper presents a numerical scheme designed for solving fractional linear optimal control problems (FLOCPs) governed by Caputo fractional derivatives (CFDs) of order 0<α≤1. The method transforms the original fractional control problem into an equivalent linear programming problem (LPP), enabling a more straightforward solution process. To evaluate the performance and accuracy of this approach, several numerical experiments were conducted using MATLAB. These experiments highlight the method’s computational efficiency and its simplicity in terms of implementation. The proposed approach offers accurate results, particularly in scenarios where traditional methods may fall short due to the complexities introduced by fractional derivatives. A detailed numerical example is presented to further illustrate the method’s effectiveness in addressing FLOCPs. The outcomes suggest that this approach is not only practical but also provides valuable insights for researchers dealing with fractional calculus in optimal control theory. This contribution is expected to be beneficial for applications in engineering and the physical sciences, where fractional-order systems play a significant role.

Exponential Quasi-Synchronization of Fractional-Order Fuzzy Cellular Neural Networks via Impulsive Control

This paper investigates the exponential quasi-synchronization of fractional-order fuzzy cellular neural networks with parameters mismatch via impulsive control. Firstly, under the framework of the generalized Caputo fractional-order derivative, a new fractional-order impulsive differential inequality is established. Secondly, based on this fractional-order impulsive differential inequality, a general criterion for the quasi-synchronization of fractional-order systems is obtained. Then, specific to the fractional-order fuzzy cellular neural network model in this paper, the criteria and error estimation of the exponential quasi-synchronization of fractional-order fuzzy cellular neural networks can be obtained. Finally, two numerical examples are given to illustrate the effectiveness of the obtained results.

Does a Fractional-Order Recurrent Neural Network Improve the Identification of Chaotic Dynamics?

This paper presents a quantitative study of the effects of using arbitrary-order operators in Neural Networks. It is based on a Recurrent Wavelet First-Order Neural Network (RWFONN), which can accurately identify several chaotic systems (measured by the mean square error and the coefficient of determination, also known as R-Squared, r2) under a fixed parameter scheme in the neural algorithm. Using fractional operators, we analyze whether the identification capabilities of the RWFONN are improved, and whether it can identify signals from fractional-order chaotic systems. The results presented in this paper show that using a fractional-order Neural Network does not bring significant advantages in the identification process, compared to an integer-order RWFONN. Nevertheless, the neural algorithm (modeled with an integer-order derivative) proved capable of identifying fractional-order dynamical systems, whose behavior ranges from periodic and multi-stable to chaotic oscillations. That is, the performances of the Neural Network model with an integer-order derivative and the fractional-order network are practically identical, making the use of fractional-order RWFONN-type networks meaningless. The results deepen the work previously published by the authors, and contribute to developing structures based on robust and generic neural algorithms to identify more than one chaotic oscillator without retraining the Neural Network.

Event-Triggered Backstepping Control of Fractional-Order Chaotic Systems with Dead Zone Via Disturbance Observer

This paper comes up with an adaptive event-triggered tracking control scheme for a kind of fractional-order strict feedback nonlinear systems with actuator dead-zone inputs and uncertain external disturbances. The presented strategy can overcome the defect in standard backstepping control scheme which leads to the “explosion of complexity” problem of virtual signals, and strengthen the transmission efficiency by an event-triggered scheme. An error compensation mechanism is presented for promoting the tracking accuracy and reducing the effect filter error. The designed controller guarantees that the tracking error converges into a small neighborhood near the origin, and all closed-loop signals are bounded. Moreover, Zeno behavior will be avoided. The numerical simulation is given to verify the accuracy and effectiveness of the designed technique.

Event-Triggered Fuzzy Adaptive Predefined-Time Control for Fractional-Order Nonlinear Systems with Time-Varying Deferred Constraints and Its Application

This paper focuses on the fuzzy adaptive predefined-time control for fractional-order nonlinear systems with time-varying deferred constraints. First, a modified dynamic surface control technique is introduced to address the problem of computational complexity exposed in the backstepping framework, and the interval type-2 fuzzy logic systems are applied to model the unknown nonlinearities of the systems. Next, a shifting function and the barrier Lyapunov function with variational barrier bounds are formulated to deal with the constraints issue. Particularly, the constraint conditions can be satisfied within a predetermined time, even if they are transgressed initially. Furthermore, a switching threshold event-triggered controller is devised to balance the control energy and communication resources. With the help of the predefined-time stability criterion, it is proven that the presented predefined-time event-triggered controller can ensure that all the signals involved in the closed-loop system are bounded and the tracking error fluctuates to a small neighborhood of the origin in a predefined-time interval. Finally, two simulation examples are provided to confirm the effectiveness of the put-forward control algorithm.

Research on fractional-order memory system signals based on Loop-By-Loop Progressive Iterative Method

This article abandons the traditional Laplace transform and proposes a new method for studying fractional-order circuits, which is the Loop-By-Loop Progressive Iterative Method(LPIM). Firstly, in order to demonstrate the correctness of LPIM, the fractance circuit, which is a relatively mature and simple form in fractional-order circuits, was chosen as the research object. The output signals of fractance circuit were studied for the first time using Laplace transform and LPIM, respectively. The results showed that the conclusions obtained by LPIM were completely consistent with those obtained by Laplace transform method and existing theories, thus verifying the correctness of LPIM. Then, a brand new Fractional-Order Memory Systems (FMS) model is constructed, and based on this model, LPIM is used for the first time to simulate the output signal of Flux-Controlled Fractional-Order Memory Systems (FFMS) that has not been studied so far. The results show that when a sine signal is used as the excitation signal, the output signal of the FFMS intersects at two points, and the output signal is modulated by the frequency of the excitation signal. Finally, combining existing theories, predict the output commonalities of FMS.

On Observer and Controller Design for Nonlinear Hadamard Fractional-Order One-Sided Lipschitz Systems

This paper presents an extensive investigation into the state feedback stabilization, observer design, and observer-based controller design for a specific category of nonlinear Hadamard fractional-order systems. The research extends the conventional integer-order derivative to the Hadamard fractional-order derivative, offering a more universally applicable method for system analysis. Furthermore, the traditional Lipschitz condition is adapted to a one-sided Lipschitz condition, broadening the range of systems amenable to analysis using these techniques. The efficacy of the proposed theoretical findings is illustrated through several numerical examples. For instance, in Example 1, we select an order of derivative r = 0.8; in Example 2, r is set to 0.9; and in Example 3, r = 0.95. This study enhances the comprehension and regulation of nonlinear Hadamard fractional-order systems, setting the stage for future explorations in this domain.

Stabilization using the separation principle for generalized classes of fractional-order fuzzy systems

Stabilization using the separation principle for generalized classes of fractional-order fuzzy systems

Modified Hermite Wavelet Discrete Matrix Approach for the Brusselator Chemical Model

AbstractThe primary goal of this study is to create a wavelet collocation technique that can be used to solve nonlinear fractional order systems of ordinary differential equations, which are equations that arise in modeling problems related to auto‐catalytic chemical reactions. Using the Hermite wavelet collocation method (HWCM), the system of nonlinear ordinary differential equations of integer and fractional order is numerically solved. The nonlinear Brusselator system is transformed into an algebraic equation system using the collocation method and the fractional derivative operational matrices. The Newton‐Raphson method is used to solve these algebraic equations, and the approximate values of the derived unknown coefficients are substituted. Through the numerical examples, the method's computational effectiveness and correctness are illustrated with various model constraints. A numerical comparison is made between the current approach ND solver, RK method, and Haar wavelet method (HWM). The efficiency and reliability of the developed strategy's performance are shown in graphs and tables. The created Hermite wavelet collocation method is resilient and has good accuracy compared to current methods found in the literature. Numerical computations are performed through Mathematica, a mathematical software.