Research

24 pages, 2680 KiB

Open AccessArticle

Observer-Based Adaptive Fuzzy Quantized Control for Fractional-Order Nonlinear Time-Delay Systems with Unknown Control Gains

by Yuwen Dong, Shuai Song, Xiaona Song and Inés Tejado

Mathematics 2024, 12(2), 314; https://doi.org/10.3390/math12020314 - 18 Jan 2024

Viewed by 566

Abstract

This paper investigates the observer-based adaptive fuzzy quantized control problem for a class of fractional-order nonlinear time-delay systems with unknown control gains based on a modified fractional-order dynamic surface control (FODSC) technique and an indirect Lyapunov method. First, a fractional-order, high-gain state observer [...] Read more.

This paper investigates the observer-based adaptive fuzzy quantized control problem for a class of fractional-order nonlinear time-delay systems with unknown control gains based on a modified fractional-order dynamic surface control (FODSC) technique and an indirect Lyapunov method. First, a fractional-order, high-gain state observer is constructed to estimate unavailable state information. Furthermore, the Nussbaum gain technique and a fractional-order filter are adopted to cope with the problem of unknown control gains and to reduce the computational complexity of the conventional recursive procedure, respectively. Moreover, through integration with the compensation mechanism and estimation model, the adaptive fuzzy quantized controllers and adaptive laws are designed to ensure that all the signals of the closed-loop system are bounded. In the end, the proposed controller is applied to a numerical example and a single-machine-infinite bus (SMIB) power system; the simulation results show the validity, superiority, and application potential of the developed control strategy. Full article

(This article belongs to the Special Issue Mathematical Methods for Nonlinear Dynamics)

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17 pages, 297 KiB

Open AccessArticle

Existence Results for Systems of Nonlinear Second-Order and Impulsive Differential Equations with Periodic Boundary

by Abdelkader Moumen, Amin Benaissa Cherif, Mohamed Ferhat, Mohamed Bouye and Khaled Zennir

Mathematics 2023, 11(24), 4907; https://doi.org/10.3390/math11244907 - 08 Dec 2023

Viewed by 599

Abstract

A class for systems of nonlinear second-order differential equations with periodic impulse action are considered. An urgent problem for this class of differential equations is the problem of the quantitative study (existence) in the case when the phase space of the equation is, [...] Read more.

A class for systems of nonlinear second-order differential equations with periodic impulse action are considered. An urgent problem for this class of differential equations is the problem of the quantitative study (existence) in the case when the phase space of the equation is, in the general case, some Banach space. In this work, sufficient conditions for the existence of solutions for a system with parameters are obtained. The results are obtained by using fixed point theorems for operators on a cone. Our approach is based on Schaefer’s fixed point theorem more precisely. In addition, the existence of positive solutions is also investigated. Full article

(This article belongs to the Special Issue Mathematical Methods for Nonlinear Dynamics)

18 pages, 3613 KiB

Open AccessArticle

Complex Dynamic Analysis for a Rent-Seeking Game with Political Competition and Policymaker Costs

by Xiuqin Yang, Feng Liu and Hua Wang

Mathematics 2023, 11(21), 4524; https://doi.org/10.3390/math11214524 - 02 Nov 2023

Viewed by 503

Abstract

This paper investigates the dynamics of rent-seeking games that include political competition and policymaker cost model. The local asymptotic stability of multiple equilibrium points and Nash equilibrium points are studied. In the rent-seeking model, the existence and stability of Flip bifurcation and Neimark–Sacker [...] Read more.

This paper investigates the dynamics of rent-seeking games that include political competition and policymaker cost model. The local asymptotic stability of multiple equilibrium points and Nash equilibrium points are studied. In the rent-seeking model, the existence and stability of Flip bifurcation and Neimark–Sacker bifurcation are examined, and the corresponding theorems and conditions are derived. The theoretical conclusions of the paper are verified by numerical simulations with different parameters. The simulation graphics show that the rent-seeking game model exhibits rich dynamic behaviors, such as multi-periodic orbits, Flip bifurcation, Neimark–Sacker bifurcation, and chaotic sets. Full article

(This article belongs to the Special Issue Mathematical Methods for Nonlinear Dynamics)

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17 pages, 1104 KiB

Open AccessArticle

Wave Propagation and Stability Analysis for Ostrovsky and Symmetric Regularized Long-Wave Equations

by Melike Kaplan, Rubayyi T. Alqahtani and Nadiyah Hussain Alharthi

Mathematics 2023, 11(19), 4030; https://doi.org/10.3390/math11194030 - 22 Sep 2023

Cited by 4 | Viewed by 644

Abstract

This work focuses on the propagation of waves on the water’s surface, which can be described via different mathematical models. Here, we apply the generalized exponential rational function method (GERFM) to several nonlinear models of surface wave propagation to identify their multiple solitary [...] Read more.

This work focuses on the propagation of waves on the water’s surface, which can be described via different mathematical models. Here, we apply the generalized exponential rational function method (GERFM) to several nonlinear models of surface wave propagation to identify their multiple solitary wave structures. We provide stability analysis and graphical representations for the considered models. Additionally, this paper compares the results obtained in this work and existing solutions for the considered models in the literature. The effectiveness and potency of the utilized approach are demonstrated, indicating their applicability to a broad range of nonlinear partial differential equations in physical phenomena. Full article

(This article belongs to the Special Issue Mathematical Methods for Nonlinear Dynamics)

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8 pages, 255 KiB

Open AccessArticle

Deriving Euler’s Equation for Rigid-Body Rotation via Lagrangian Dynamics with Generalized Coordinates

by Dennis S. Bernstein, Ankit Goel and Omran Kouba

Mathematics 2023, 11(12), 2727; https://doi.org/10.3390/math11122727 - 16 Jun 2023

Cited by 1 | Viewed by 2162

Abstract

Euler’s equation relates the change in angular momentum of a rigid body to the applied torque. This paper uses Lagrangian dynamics to derive Euler’s equation in terms of generalized coordinates. This is done by parameterizing the angular velocity vector in terms of 3-2-1 [...] Read more.

Euler’s equation relates the change in angular momentum of a rigid body to the applied torque. This paper uses Lagrangian dynamics to derive Euler’s equation in terms of generalized coordinates. This is done by parameterizing the angular velocity vector in terms of 3-2-1 and 3-1-3 Euler angles as well as Euler parameters, that is, quaternions. This paper fills a gap in the literature by using generalized coordinates to parameterize the angular velocity vector and thereby transform the dynamics obtained from Lagrangian dynamics into Euler’s equation for rigid-body rotation. Full article

(This article belongs to the Special Issue Mathematical Methods for Nonlinear Dynamics)

19 pages, 2517 KiB

Open AccessArticle

Grey-Black Optical Solitons, Homoclinic Breather, Combined Solitons via Chupin Liu’s Theorem for Improved Perturbed NLSE with Dual-Power Law Nonlinearity

by Syed T. R. Rizvi, Aly R. Seadawy and Shami A. M. Alsallami

Mathematics 2023, 11(9), 2122; https://doi.org/10.3390/math11092122 - 30 Apr 2023

Cited by 3 | Viewed by 875

Abstract

In this article, we consider the improved perturbed nonlinear Schrödinger Equation (IP-NLSE) with dual power law nonlinearity, which arises in optical fibers and photovoltaic-photo-refractive materials. We found grey and black optical solitons of the governing equation by means of a suitable complex envelope [...] Read more.

In this article, we consider the improved perturbed nonlinear Schrödinger Equation (IP-NLSE) with dual power law nonlinearity, which arises in optical fibers and photovoltaic-photo-refractive materials. We found grey and black optical solitons of the governing equation by means of a suitable complex envelope ansatz solution. By using the Chupin Liu’s theorem (CLT) for the grey and black solitons, we evaluated new categories of combined optical soliton (COS) solutions to the IP-NLSE. The propagation behaviors for homoclinic breathers (HB), multiwaves and M-shape solitons will be analytically examined. All new analytical solutions will be found by an ansatz function scheme and suitable transformations. Multiwave solitons have been reported by using a three-waves technique. Furthermore, two kinds of interactions for M-shape soliton through exponential functions will be examined. Full article

(This article belongs to the Special Issue Mathematical Methods for Nonlinear Dynamics)

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22 pages, 1688 KiB

Open AccessArticle

Three-Species Predator–Prey Stochastic Delayed Model Driven by Lévy Jumps and with Cooperation among Prey Species

by Jaouad Danane and Delfim F. M. Torres

Mathematics 2023, 11(7), 1595; https://doi.org/10.3390/math11071595 - 25 Mar 2023

Cited by 1 | Viewed by 1116

Abstract

Our study focuses on analyzing the behavior of a stochastic predator–prey model with a time delay and logistic growth of prey, influenced by Lévy noise. Initially, we establish the existence, uniqueness, and boundedness of a positive solution that spans globally. Subsequently, we explore [...] Read more.

Our study focuses on analyzing the behavior of a stochastic predator–prey model with a time delay and logistic growth of prey, influenced by Lévy noise. Initially, we establish the existence, uniqueness, and boundedness of a positive solution that spans globally. Subsequently, we explore the conditions under which extinction occurs, and identify adequate criteria for persistence. Finally, we validate our theoretical findings through numerical simulations, which also helps illustrate the dynamics of the stochastic delayed predator–prey model based on different criteria. Full article

(This article belongs to the Special Issue Mathematical Methods for Nonlinear Dynamics)

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16 pages, 604 KiB

Open AccessArticle

Adaptive Hyperparameter Fine-Tuning for Boosting the Robustness and Quality of the Particle Swarm Optimization Algorithm for Non-Linear RBF Neural Network Modelling and Its Applications

by Zohaib Ahmad, Jianqiang Li and Tariq Mahmood

Mathematics 2023, 11(1), 242; https://doi.org/10.3390/math11010242 - 03 Jan 2023

Cited by 8 | Viewed by 1776

Abstract

A method is proposed for recognizing and predicting non-linear systems employing a radial basis function neural network (RBFNN) and robust hybrid particle swarm optimization (HPSO) approach. A PSO is coupled with a spiral-shaped mechanism (HPSO-SSM) to optimize the PSO performance by mitigating its [...] Read more.

A method is proposed for recognizing and predicting non-linear systems employing a radial basis function neural network (RBFNN) and robust hybrid particle swarm optimization (HPSO) approach. A PSO is coupled with a spiral-shaped mechanism (HPSO-SSM) to optimize the PSO performance by mitigating its constraints, such as sluggish convergence and the local minimum dilemma. Three advancements are incorporated into the hypothesized HPSO-SSM algorithms to achieve remarkable results. First, the diversity of the search process is promoted to update the inertial weight

ω

based on the logistic map sequence. Then, two distinct parameters are trained in the original position update algorithm to enhance the work efficiency of the successive generation. Finally, the proposed approach employs a spiral-shaped mechanism as a local search operator inside the optimum solution space. Moreover, the HPSO-SSM method concurrently improves the RBFNN parameters and network size, building a model with a compact configuration and higher precision. Two non-linear benchmark functions and the total phosphorus (TP) modelling issue in a waste water treatment process (WWTP) are utilized to assess the overall efficacy of the creative technique. The results of testing indicate that the projected HPSO-SSM-RBFNN algorithm performed very effectively. Full article

(This article belongs to the Special Issue Mathematical Methods for Nonlinear Dynamics)

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15 pages, 1679 KiB

Open AccessArticle

Improved Confidence Interval and Hypothesis Testing for the Ratio of the Coefficients of Variation of Two Uncorrelated Populations

by Abbas Bahrampour, Zeynab Avazzadeh, Mohammad Reza Mahmoudi and António M. Lopes

Mathematics 2022, 10(19), 3495; https://doi.org/10.3390/math10193495 - 25 Sep 2022

Viewed by 1219

Abstract

One of the most accessible and useful statistical tools for comparing independent populations in different research areas is the coefficient of variation (CV). In this study, first, the asymptotic distribution of the ratio of CV of two uncorrelated populations is investigated. Then, the [...] Read more.

One of the most accessible and useful statistical tools for comparing independent populations in different research areas is the coefficient of variation (CV). In this study, first, the asymptotic distribution of the ratio of CV of two uncorrelated populations is investigated. Then, the outputs are used to create a confidence interval and to establish a test of hypothesis about the CV ratio of the populations. The proposed approach is compared with an alternative method, showing its superiority and effectiveness. Full article

(This article belongs to the Special Issue Mathematical Methods for Nonlinear Dynamics)

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18 pages, 394 KiB

Open AccessArticle

Numerical Analysis of Alternating Direction Implicit Orthogonal Spline Collocation Scheme for the Hyperbolic Integrodifferential Equation with a Weakly Singular Kernel

by Qiong Huang, Omid Nikan and Zakieh Avazzadeh

Mathematics 2022, 10(18), 3390; https://doi.org/10.3390/math10183390 - 19 Sep 2022

Cited by 1 | Viewed by 1173

Abstract

This paper studies an alternating direction implicit orthogonal spline collocation (ADIOSC) technique for calculating the numerical solution of the hyperbolic integrodifferential problem with a weakly singular kernel in the two-dimensional domain. The integral term is approximated with the help of the second-order fractional [...] Read more.

This paper studies an alternating direction implicit orthogonal spline collocation (ADIOSC) technique for calculating the numerical solution of the hyperbolic integrodifferential problem with a weakly singular kernel in the two-dimensional domain. The integral term is approximated with the help of the second-order fractional quadrature formula introduced by Lubich. The stability and convergence analysis of the proposed strategy are proven in

L^{2}

-norm. Numerical results highlight the high accuracy and efficiency of the proposed strategy and clarify the theoretical prediction. Full article

(This article belongs to the Special Issue Mathematical Methods for Nonlinear Dynamics)

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9 pages, 1362 KiB

Open AccessArticle

Extended Tanh-Function Method and Its Applications in Nonlocal Complex mKdV Equations

by Xiaodong Wang, Jianping Wu, Yazi Wang and Can Chen

Mathematics 2022, 10(18), 3250; https://doi.org/10.3390/math10183250 - 07 Sep 2022

Cited by 1 | Viewed by 1098

Abstract

In order to construct the multiple traveling wave solutions of the nonlocal modified Korteweg de Vires (mKdV) equation, the modified tanh-function approach for local soliton equations is extended to a nonlocal complex mKdV equation. The central idea of this method is to use [...] Read more.

In order to construct the multiple traveling wave solutions of the nonlocal modified Korteweg de Vires (mKdV) equation, the modified tanh-function approach for local soliton equations is extended to a nonlocal complex mKdV equation. The central idea of this method is to use the solution of the Riccati equation to replace the tanh function in the tanh function (THF) method. As an application, we investigate a new traveling wave solution for the nonlocal complex mKdV equation of Ablowitz and Musslimani. Moreover, some exciting diagrams show the underlying dynamics of some given solutions. Full article

(This article belongs to the Special Issue Mathematical Methods for Nonlinear Dynamics)

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20 pages, 957 KiB

Open AccessArticle

High-Order Multivariate Spectral Algorithms for High-Dimensional Nonlinear Weakly Singular Integral Equations with Delay

by Ahmed Z. Amin, Mahmoud A. Zaky, Ahmed S. Hendy, Ishak Hashim and Ahmed Aldraiweesh

Mathematics 2022, 10(17), 3065; https://doi.org/10.3390/math10173065 - 25 Aug 2022

Cited by 4 | Viewed by 1077

Abstract

One of the open problems in the numerical analysis of solutions to high-dimensional nonlinear integral equations with memory kernel and proportional delay is how to preserve the high-order accuracy for nonsmooth solutions. It is well-known that the solutions to these equations display a [...] Read more.

One of the open problems in the numerical analysis of solutions to high-dimensional nonlinear integral equations with memory kernel and proportional delay is how to preserve the high-order accuracy for nonsmooth solutions. It is well-known that the solutions to these equations display a typical weak singularity at the initial time, which causes challenges in developing high-order and efficient numerical algorithms. The key idea of the proposed approach is to adopt a smoothing transformation for the multivariate spectral collocation method to circumvent the curse of singularity at the beginning of time. Therefore, the singularity of the approximate solution can be tailored to that of the exact one, resulting in high-order spectral collocation algorithms. Moreover, we provide a framework for studying the rate of convergence of the proposed algorithm. Finally, we give a numerical test example to show that the approach can preserve the nonsmooth solution to the underlying problems. Full article

(This article belongs to the Special Issue Mathematical Methods for Nonlinear Dynamics)

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