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Mathematical Methods for Nonlinear Dynamics
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Mathematical Methods for Nonlinear Dynamics

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Mathematical Physics".

Deadline for manuscript submissions: closed (31 July 2024) | Viewed by 18158

Special Issue Editors

Dr. António Lopes

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Guest Editor
Faculty of Engineering, University of Porto, Rua Dr. Roberto Frias, 4200–465 Porto, Portugal
Interests: complex systems modelling; automation and robotics; fractional order systems modelling and control; data analysis and visualization
Special Issues, Collections and Topics in MDPI journals
Dr. Omid Nikan

E-Mail Website
Guest Editor
School of Mathematics, Iran University of Science and Technology, Narmak, Tehran 16846-13114, Iran
Interests: fractional calculus; PDE; nonlinear dynamic; computational techniques
Special Issues, Collections and Topics in MDPI journals
Dr. Zakieh Avazzadeh

E-Mail Website
Guest Editor
Department of Applied Mathematics, Xi'an Jiaotong-Liverpool University, Suzhou 215123, China
Interests: fractional calculus; PDE; optimal control; nonlinear dynamics; numerical approximation method
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

The modeling and control of nonlinear dynamic systems can be challenging in mathematics and engineering. Despite much investigation carried out so far, nonlinear and complex phenomena are not fully understood yet, due to their considerable randomness and a diversity of reasons underlying the energy dissipation involving the dynamic effects. In fact, present knowledge and scientific tools are still far from capturing the overall richness of the systems dynamics.

The Special Issue focuses on the mathematical modeling and control of nonlinear dynamical systems. Manuscripts centered around mathematical methods, analytical and numerical algorithms, advanced control systems, complex dynamics, fractional calculus and its applications, fractals and chaos, multibody systems, modelling, and simulation, among others, are welcome.

The Special Issue will collate various contributions from researchers across different topics including engineering, mathematics, physics, biology, geophysics, and other sciences. Papers describing original theoretical research as well as new experimental results are expected.

Dr. António Lopes
Dr. Omid Nikan
Dr. Zakieh Avazzadeh
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • nonlinear dynamics
  • complexity and complex systems
  • advanced control systems
  • fractional calculus
  • mathematical modelling
  • analytical and numerical methods
  • computational techniques and numerical simulation
  • signal processing
  • time series analysis
  • complex networks
  • electro-mechanical structures
  • finance end economy systems
  • biomathematics
  • evolutionary computing
  • chaos and fractals

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Published Papers (12 papers)

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Research

24 pages, 2680 KiB  
Article
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
Cited by 1 | Viewed by 780
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  
Article
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 - 8 Dec 2023
Viewed by 743
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  
Article
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 - 2 Nov 2023
Viewed by 724
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  
Article
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 5 | Viewed by 869
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  
Article
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 2787
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  
Article
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 1113
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  
Article
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 2 | Viewed by 1468
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  
Article
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 - 3 Jan 2023
Cited by 11 | Viewed by 2207
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  
Article
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 1534
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  
Article
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 2 | Viewed by 1419
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 L2-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  
Article
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 - 7 Sep 2022
Cited by 5 | Viewed by 1372
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  
Article
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 1313
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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