Nonlinear Fractional Systems Research Articles

Numerical and graphical simulation of the non-linear fractional dynamical system of bone mineralization.

The objective of the present study was to improve our understanding of the complex biological process of bone mineralization by performing mathematical modeling with the Caputo-Fabrizio fractional operator. To obtain a better understanding of Komarova's bone mineralization process, we have thoroughly examined the boundedness, existence, and uniqueness of solutions and stability analysis within this framework. To determine how model parameters affect the behavior of the system, sensitivity analysis was carried out. Furthermore, the fractional Adams-Bashforth method has been used to carry out numerical and graphical simulations. Our work is significant owing to its comparison of fractional- and integer-order models, which provides novel insight into the effectiveness of fractional operators in representing the complex dynamics of bone mineralization.

Resonance analysis and time-delay feedback controllability for a fractional horizontal nonlinear roller system

<p>In this paper, we investigated the nonlinear vibration characteristics and time-delay feedback controllability of a fractional horizontal roll system, which is described by a fractional Duffing-van der Pol oscillator under an external harmonic excitation. We focused on the resonance of fractional roller systems and conducted corresponding vibration control. The amplitude-frequency equations of primary resonance and superharmonic resonance were obtained using the multiple scale method. The amplitude-frequency characteristic curves of the system with different parameters were presented, and the influence of system parameters on the curves was analyzed. In addition, the time-delay feedback controller was designed to control the parameter excitation vibration. The numerical simulation results have verified the effectiveness of the time-delay controller in eliminating the jumping and hysteresis phenomena of the rolling system. The comparisons of approximate analytical solution and numerical solution was fulfilled, and the result certifies the correctness and satisfactory precision of the approximately analytical solution. The analysis results provide certain theoretical guidance for the vibration reduction of the horizontal nonlinear roller system.</p>

Stability of Caputa–Katugampola fractional differential nonlinear control system with delay Riemann−Katugampola

Stability of Caputa–Katugampola fractional differential nonlinear control system with delay Riemann−Katugampola

Modulating functions based state estimator for Caputo fractional systems

This paper aims to design a non-asymptotic and robust state estimator applicable both for Caputo fractional linear and nonlinear systems with noisy outputs. For this purpose, a fractional observable canonical form is considered to show the idea of the proposed method. First, the considered form is transformed by taking a fractional derivative operator. Second, based on the new form, the modulating functions method with the additive index law of fractional derivatives is used to get exact formulas for the sought variables in continuous noise-free case, which only contain the integrals involving the output and the fractional derivative of the input, without producing any source of errors. Moreover, the integral forms can reduce noisy effect as low-pass filters. Hence, fast convergent and robust estimation can be obtained using discrete noisy outputs. After constructing the required modulating functions, two numerical examples are given to demonstrate the advantages of the proposed estimator by comparing with the fractional order Luenberger-like observer.

Synchronization of generalized fractional complex networks with partial subchannel losses

<abstract><p>This article focuses on the synchronization problem for two classes of complex networks with subchannel losses and generalized fractional derivatives. Initially, a new stability theorem for generalized fractional nonlinear system is formulated using the properties of generalized fractional calculus and the generalized Laplace transform. This result is also true for classical fractional cases. Subsequently, synchronization criteria for the generalized fractional complex networks are attained by the proposed stability theorem and the state layered method. Lastly, two numerical examples with some new kernel functions are given to validate the synchronization results.</p></abstract>

On chaos control of nonlinear fractional Newton-Leipnik system via fractional Caputo-Fabrizio derivatives

In this work, we present a design for a Newton-Leipnik system with a fractional Caputo-Fabrizio derivative to explain its chaotic characteristics. This time-varying fractional Caputo-Fabrizio derivative approach is applied to solve the model numerically, and to check the solution’s existence and uniqueness. The existence and uniqueness of results of a fractional-order model under the Caputo-Fabrizio fractional operator have been proved by fixed point theory. As well, we achieved a stable result by applying the Ulam-Hyers concept. Chaos is controlled by linear controllers. Furthermore, the Lyapunov exponent of the system indicates that the chaos control findings are accurate. Based on weighted covariant Lyapunov vectors we construct a background covariance matrix using the Kaplan-Yorke dimension. Using a numerical example, this suggested method is illustrated for its applicability and efficiency.

New results on controllability analysis of nonlinear fractional order integrodifferential Langevin system with multiple delays

Nowadays, the study of fractional differential equations (FDEs) is one of the most interesting research topics. In this article, we study a novel nonlinear fractional order Langevin system involving delays. The primary focus of this article is to analyze the relative controllability of a nonlinear integrodifferential fractional Langevin dynamical system with multiple delays in control for finite-dimensional spaces. Fundamentals of the fixed-point theory are used to achieve the desired results. Using Schauder’s fixed theorem, sufficient conditions for the controllability are established and the Mittag-Leffler matrix function defines the controllability Grammian matrix. Additionally, the derived results are validated through an application.

An analysis on the optimal control and approximate controllability for Hilfer fractional neutral integro‐differential systems with finite delay

AbstractThe existence and uniqueness of solutions to Hilfer fractional neutral delay integro‐differential equations subject to nonlocal conditions are discussed in this study. The main results of this study are based in part on the fixed point techniques of Banach contraction and Krasnoselskii's fixed point theorem from calculus theory. First, we determine whether or not the fractional system has a mild solution. The uniqueness of the mild solution is further illustrated by expanding our results. The optimal control problems are governed by a new class of neutral delay integro‐differential equations in Banach spaces, and we also develop sufficient conditions for the approximate controllability of a nonlinear fractional system. An example is given at the end to strengthen the compatibility of the results.

Numerical analysis with a class of trigonometric functions for nonlinear time fractional Wu-Zhang system

We study the implementation of a numerical method to solve the time fractional (2+1)-dimensional Wu–Zhang system focusing on properties of trigonometric basis functions (TBFs). The approximation of the system describing dispersive long waves by appropriate discretization of the system and Grünwald difference operators is discussed and a linear system of the resulting discrete systems is given. This process includes a presentation of a general framework to approximate the solution by a mesh-free scheme which is stable and converges. Finally, various numerical experiments are present to illustrate the exceptional effect of fractional-order on the WZ system.

Estimation of Faults in Fractional Nonlinear Systems via Fractional Observers

In this work three fractional observers are implemented in order to estimate faults of a fractional nonlinear system and a Bergman model which gives us a real application model, the faults considered are additive faults. In this work observer-based diagnosis is implemented and we compare the fractional observers with a criterion of a performance index function. Based on this performance index, the fractional observers are compares, and is verified which one gives the best performance. We characterized three fractional observers, selecting the observer gains in order to give the best performance of each observer.

Nonlocal p-Laplacian involving a nonlinear fractional reaction-diffusion system applied to image restoration

Image restoration has been a longstanding challenge in image processing, involving the enhancement of image content to extract valuable information. Nonlinear models have demonstrated their efficacy in eliminating additive noise, which motivates our goal of investigating a novel nonlocal nonlinear reaction-diffusion model for noise removal. The proposed model integrates a fractional diffusion equation with a nonlinear nonlocal p-Laplace operator, with the fidelity term employing a weak norm to better capture oscillatory patterns and intricate details in textured images. Using the Schauder fixed point theorem, the well-posedness of the proposed model solution is established. The experimental results confirm the effectiveness and efficiency of the proposed model, providing validation for its practical utility.

Controllability of fractional dynamical systems having multiple delays in control with ψ‐Caputo fractional derivative

This study investigates the controllability problems for linear and nonlinear fractional control systems with multiple control delays in the context of the ‐Caputo fractional derivative. For the linear case, using the positive definiteness of the Gramian matrix, the necessary and sufficient conditions are obtained. For the nonlinear case, utilizing the iterative technique together with the completeness of the space, sufficient conditions for the existence of a solution are obtained. Just a few examples are included to make theoretical findings easier to grasp, and their respective graphs are plotted.

Non-existence Results for a Nonlinear Fractional System of Differential Problems

Non-existence Results for a Nonlinear Fractional System of Differential Problems

LMI Based Synchronization Control of Nonlinear Affine Fractional Order Chaotic Systems Considering Input Constraint

LMI Based Synchronization Control of Nonlinear Affine Fractional Order Chaotic Systems Considering Input Constraint

A New Extension of Optimal Auxiliary Function Method to Fractional Non-Linear Coupled ITO System and Time Fractional Non-Linear KDV System

In this article, we investigate the utilization of Riemann–Liouville’s fractional integral and the Caputo derivative in the application of the Optimal Auxiliary Function Method (OAFM). The extended OAFM is employed to analyze fractional non-linear coupled ITO systems and non-linear KDV systems, which feature equations of a fractional order in time. We compare the results obtained for the ITO system with those derived from the Homotopy Perturbation Method (HPM) and the New Iterative Method (NIM), and for the KDV system with the Laplace Adomian Decomposition Method (LADM). OAFM demonstrates remarkable convergence with a single iteration, rendering it highly effective. In contrast to other existing analytical approaches, OAFM emerges as a dependable and efficient methodology, delivering high-precision solutions for intricate problems while saving both computational resources and time. Our results indicate superior accuracy with OAFM in comparison to HPM, NIM, and LADM. Additionally, we enhance the accuracy of OAFM through the introduction of supplementary auxiliary functions.

Results on approximate controllability for fractional stochastic delay differential systems of order r ∈ (1,2)

In this paper, we deal with the approximate controllability of fractional stochastic delay differential inclusions of order [Formula: see text]. By using fractional calculus, stochastic analysis, the theory of cosine family and Dhage fixed point techniques, a new set of necessary and sufficient conditions are formulated which guarantees the approximate controllability of the nonlinear fractional stochastic system. In particular, the results are established with the assumption that the associated linear part of the system is approximately controllable. Further, the result is extended to obtain the conditions for the solvability of controllability results for fractional inclusions with nonlocal conditions. Finally, an example is presented to illustrate the theory of the obtained result.

Robust optimal control of nonlinear fractional systems

Parameter uncertainty is inevitably found in almost every system and, if is ignored, it could jeopardize the effectiveness of control method. Motivated by this issue, we consider a robust optimal control problem governed by a nonlinear fractional system with uncertain parameters, where the system sensitivity with respect to the uncertain parameters is explicitly included in the cost functional. For this problem, we first prove that the system sensitivity can be expressed as the solution of an auxiliary fractional system. Then, we propose a numerical scheme for discretizing both the original and auxiliary fractional systems, resulting in a finite-dimensional optimization problem. Furthermore, a numerical solution algorithm based on gradients of the cost functional is developed for the resulting optimization problem. Finally, numerical results for two example problems are given to demonstrate the validity and applicability of the proposed algorithm.

Controllability of nonlinear fractional Langevin systems using $${\varPsi }$$-Caputo fractional derivative

Controllability of nonlinear fractional Langevin systems using $${\varPsi }$$-Caputo fractional derivative

Noise-boosted weak signal detection in fractional nonlinear systems enhanced by increasing potential-well width and its application to mechanical fault diagnosis

Noise is seen as the annoying something for weak signal detection, but noise is beneficial to enhance weak signals of interest embedded in noise in nonlinear systems. Moreover, the fractional-order derivative can reinforce noise-boosted weak signal detection. However, the role of varying potential-well depth and width individually on noise-boosted weak signal detection in overdamped and underdamped fractional-order nonlinear systems has not been investigated yet. For this purpose, this paper designs a special bistable potential with varying potential-well depth and width individually to study their influences on weak signal detection numerically. Then, based on simulated conclusions a noise-boosted weak fault diagnosis method enhanced by increasing potential-well width is proposed to enhance weak fault characteristics of machinery, where the amplitude amplification factor is seen as an indicator to quantify the performance of weak signal detection. Finally, some simulations and a bearing fault experiment with an outer race defect were performed to demonstrate the proposed method. Simulated conclusions are identical to experimental ones and indicate that increasing potential-well width would improve noise-boosted weak signal detection but increasing potential-well depth would weaken it regardless of underdamped or overdamped fractional-order nonlinear systems. Nonetheless, increasing the potential-well width endlessly would make the amplitude at the outer race fault characteristic frequency unchanged. That is because the signal with the constant energy would not accelerate the particles to jump across the wide potential wells adequately, where it is not an optimal SR. The compared results with a classical denoising method demonstrate the superiority of the proposed method further.

A Study on Existence and Controllability of Conformable Impulsive Equations

We study the existence/uniqueness of conformable fractional type impulsive nonlinear systems as well as the controllability of linear/semilinear conformable fractional type impulsive controlled systems. Using the conformable fractional derivative approach, we introduce the conformable controllability operator and the conformable controllability Gramian matrix in order to obtain the necessary and sufficient conditions for the complete controllability of linear impulsive conformable systems. We present a set of sufficient conditions for the controllability of the conformable semilinear impulsive systems.