Fractional-order Systems Research Articles

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.

Efficacy of Continued Fraction Expansion technique in the approximation of fractional order systems

Efficacy of Continued Fraction Expansion technique in the approximation of fractional order systems

A new chemical networked system: spatial-temporal evolution and control

Abstract This paper constructs a scale-free chemical network based on the Gierer-Meinhardt (GM) system and investigates its Turing instability. We establish a fractional-order single-node GM system with delay and design a fractional-order proportional derivative (PD) control strategy for the issue of bifurcation control. Using delay as bifurcation parameter, the existence of Hopf bifurcation is proven, and control over bifurcation threshold points is achieved through a fractional-order PD control strategy. For the scale-free chemical network based on the GM system, we obtain the condition of how the Turing instability occurs. We derive how the number of edges for the new nodes changes the stability of the network-organized system and investigate the relationship between degrees of nodes and eigenvalues of the network matrix. We give the instability condition caused by diffusion in the network-organized system. Finally, the numerical simulations verify analytical results.

New modifications of natural transform iterative method and q-homotopy analysis method applied to fractional order KDV-Burger and Sawada–Kotera equations

This manuscript presents enhanced versions of two methods: the natural transform iterative method (NTIM) and the q-homotopy analysis method (q-HAM). These methods harness concepts from Fractional Calculus, particularly leveraging the Caputo fractional derivative operator, to successfully manage the complexities of fractional-order systems. To validate their accuracy and efficiency, we applied the proposed techniques to FPDEs like the fractional-order KDV-Burger and fifth-order Sawada–Kotera equations. Our outcomes, which closely resemble the exact solutions, demonstrate how useful NTIM and q-HAM are for solving difficult FPDEs and improving the study of fractional calculus.

Finite Time Stability Analysis and Feedback Control for Takagi–Sugeno Fuzzy Time Delay Fractional-Order Systems

This study treats the problem of Finite Time Stability Analysis (FTSA) and Finite Time Feedback Control (FTFC), using a Linear Matrix Inequalities Approach (LMIA). It specifically focuses on Takagi–Sugeno fuzzy Time Delay Fractional-Order Systems (TDFOS) that include nonlinear perturbations and interval Time Varying Delays (ITVDs). We consider the case of the Caputo Tempered Fractional Derivative (CTFD), which generalizes the Caputo Fractional Derivative (CFD). Two main results are presented: a two-step procedure is provided, followed by a more relaxed single-step procedure. Two examples are presented to show the reduction in conservatism achieved by the proposed methods. The first is a numerical example, while the second involves the FTFC of an inverted pendulum, which exhibits both symmetrical dynamics for small angular displacements and asymmetrical dynamics for larger deviations.

Adaptive event-triggered stochastic estimator-based sampled-data fuzzy control for fractional-order permanent magnet synchronous generator-based wind energy systems

This study aims to develop an adaptive event-triggered estimation sampled-data control method for a fractional-order permanent magnet synchronous generator (PMSG)-based wind energy system (WES) using the Takagi–Sugeno (T–S) fuzzy approach. Unlike existing PMSG-based WES control schemes, we propose an aperiodic event-triggered communication scheme with an adaptive mechanism to reduce data transmission. To address the challenge of limited communication resources, we introduce an adaptive event-triggered (AET) estimation method with aperiodic sampling, significantly reducing communication overhead while maintaining stable WES performance. This method employs transmission techniques based on absolute errors, resulting in high data transmission efficiency. First, the fractional-order model for the PMSG-based WES is represented using linear sub-models through the T–S fuzzy approach. Next, a fuzzy aperiodic AET mechanism is proposed for the PMSG model with augmented estimation error systems. Then, the fractional Lyapunov function theory is employed to derive linear matrix inequalities (LMIs), ensuring bounded mean-square stability for the PMSG-based WES with the augmented estimation error systems. Additionally, the desired estimator control gains are determined through solvable LMIs. Finally, simulation studies are presented to demonstrate the superiority and feasibility of the proposed control scheme.

Analysis and dynamical transmission of tuberculosis model with treatment effect by using fractional operator

ABSTRACT Each year, millions of people die from the airborne infectious illness tuberculosis (TB). Several drug-susceptible (DS) and drug-resistant (DR) forms of the causative agent, Mycobacterium tuberculosis (MTB), are currently common in the majority of affluent and developing nations, particularly in Bangladesh, and completely drug-resistant strains are beginning to arise. The main purpose of this research is to develop and examine a non-integer-order mathematical model for the dynamics of tuberculosis transmission using the fractal fractional operator. By demonstrating characteristics such as the boundedness of solutions, positivity, and reliance of the solution on the original data, the biological well-posedness of the mathematical model formulation was investigated for TB cases from 2002 to 2017 in KPK Pakistan. Ulam-Hyres stability is also used to assess both local and global aspects of TB bacterial infection. Sensitivity analysis of the TB model with therapy was also examined. The advanced numerical technique is used to find the solution of the fractional-order system to check the impact of fractional parameters. Simulation highlights that all classes have converging qualities and retain established positions with time, which shows the actual behavior of bacterial infection with TB.

A new method for converting impulsive Riemann–Liouville fractional order system into the integral equation

A new method for converting impulsive Riemann–Liouville fractional order system into the integral equation

Hamilton energy, competitive modes and ultimate bound estimation of a new 3D chaotic system, and its application in chaos synchronization

Abstract This paper discusses the dynamical behavior of a new 3D chaotic system of integer and fractional order. To get a comprehensive knowledge of the dynamics of the proposed system, we have studied competitive modes and Hamilton energy for different parameter values. In order to get the ultimate bound set for the proposed system, we employed the Lagrange coefficient approach to solve the optimization problem. We have also explored the use of the bound set in synchronization. Furthermore, we have examined the Hamilton energy, time series, bifurcation diagrams, and Lyapunov exponents for the fractional version of the proposed chaotic system. Finally, we looked at the Mittage-Leffler positive invariant sets and global attractive sets by merging the Lyapunov function approach with the Mittage-Leffler function. Numerical simulations have shown the obtained bound sets and other analytical outcomes.

Exploring the dynamic behavior of system model construction with unknown parameters and achieving synchronous control of fuzzy neural networks

This study proposes a method that combines constant terms with state variables to construct the target system model. The three-dimensional system model is used for verification, starting with integer-order and fractional-order continuous system models. Through the integration of information entropy, path entropy, permutation entropy, the SALI algorithm, and complexity methods, the dynamic behavior of the target model is thoroughly analyzed. Finally, by considering the range of values when the constant term is involved in the construction, the study integrates fuzzy neural networks to effectively construct a self-organizing fuzzy neural network with parameters(SFNNP). Using a three-dimensional continuous system as a simulation case, the study successfully realizes the synchronous control of a fuzzy neural network based on fuzzy parameterization.

Fractional impulsive controller design of fractional-order fuzzy systems with average dwell-time strategy and its application to wind energy systems

In this study, the issue of adaptive impulsive control of a permanent magnet synchronous generator (PMSG)-based wind energy system (WES) with an average dwell-time strategy is investigated using a Takagi–Sugeno (T-S) fuzzy model described by fractional-order differential equations. A T-S fuzzy model is employed to describe the nonlinear fractional-order PMSG (FOPMSG) model, an average dwell time, an average impulsive condition, a Lyapunov function, and a less conservative algebraic inequality criterion that guarantees stabilization for the considered nonlinear system. First, the mathematical model of the FOPMSG in the n−m reference frame is transformed into a dimensionless chaotic system through affine transformation and time scale transformation. Then, the stabilization problem of the nonlinear chaotic FOPMSG model is considered to validate the proposed sufficient conditions, leading to the derivation of a new stability criterion for the FOPMSG model, instead of addressing the general problem to validate the proposed result. Subsequently, the desired control gains can be obtained to ensure the stabilization of the addressed closed-loop system. By combining adaptive and impulsive control, the model under consideration can be stabilized for any target dynamics. Finally, we demonstrate the efficiency and feasibility of the suggested approach through numerical simulations and comparative results.

Sliding mode control design using a generalized reduced-order fractional model for chemical processes

This article presents a comprehensive study that evaluates the potential advantages of adopting a unified, systematic, and organized general fractional model within the Sliding Mode Control (SMC) design framework. The exploration extends to the model's applicability in a variety of disciplines. Although prior research has utilized SMC and fractional-order systems independently, no previous work has established a mathematical framework that integrates a generic fractional-order SMC model. The study addresses this gap and highlights potential implications for control design and applications in a variety of fields, particularly for chemical processes.

A three-dimensional discrete fractional-order HIV-1 model related to cancer cells, dynamical analysis and chaos control

In this paper, we study a three-dimensional discrete-time model to describe the behavior of cancer cells in the presence of healthy cells and HIV-infected cells. Based on the Caputo-like difference operator, we construct the fractional-order biological system. This study's significance lies in developing a new approach to presenting a biological dynamical system. Since the qualitative analysis related to existence, uniqueness, and stability is almost the same as can be found in numerous existing papers, and comparing this study to other research, constructing a biological discrete system using the Caputo difference operator can be particularly important. Using powerful tools of nonlinear theory such as phase plots, bifurcation diagrams, Lyapunov exponent spectrum, and the 0-1 test, we establish that the proposed system can exhibit different biological states, including stable, periodic, and chaotic behaviors. Here, the route leading to chaos is period-doubling bifurcation. Furthermore, the level of chaos in the system is quantified using $C_{0}$ complexity and approximate entropy algorithms. The stabilization or suppression of chaotic motions in the fractional-order system is presented, where an efficient controller is designed based on the stability theory of the discrete-time fractional-order systems. Numerical simulations are provided to validate the theoretical results derived in this research paper.

Birth of Strange Non-chaotic Attractors in Fractional-Order Systems

In this study, we present the construction of a quasiperiodically driven fractional-order Chua system and investigate its dynamical behavior by varying the fractional order parameters q1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$q_1$$\\end{document} and q2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$q_2$$\\end{document} using various characterization techniques. Our observations suggest that selecting appropriate combinations of fractional orders allows us to observe bifurcation phenomena for different values of the forcing amplitude. Through the use of Poincaré sections, we visualize the formation of tori and their doubling bifurcation route to strange nonchaotic attractor. Additionally, we employ numerical and statistical tools, such as singular continuous spectra, separation of nearby trajectories, and fast Fourier transform, to verify the occurrence and understand the mechanism behind the birth of strange nonchaotic attractor. We differentiate torus, strange nonchaos, and chaos with the help of 0–1 test and recurrence analysis. Our results provide valuable insights into the dynamical behavior of quasiperiodically driven fractional-order systems, with potential implications for various fields of science and engineering.