Topic Editors

Department of Mathematics, Texas A and M University-Kingsville, Kingsville, TX 78363, USA
Dipartimento di Matematica e Informatica, University of Catania, 95124 Catania, CT, Italy

Advances in Nonlinear Dynamics: Methods and Applications

Abstract submission deadline
20 August 2024
Manuscript submission deadline
20 October 2024
Viewed by
63297

Topic Information

Dear Colleagues,

In this Topic we shall focus on partial differential equations and their applications, especially those arising in the environmental, physical, and engineering sciences. In particular, this Special Issue will cover topics such as initial and boundary value problems, Navier–Stokes theory, minimizers for functionals of double phase with variable exponents, magnetohydrodynamics equations, Lie groups, mathematical modeling, analysis for traveling waves, and Boussinesq equations. All manuscripts are expected to be written for a broad scientific audience.

The scope of this Topic is to bring together theories, methods, and real-world applications of Nonlinear Dynamics. It will consist of topical research including (but not limited to) the following areas:

  • Ordinary differential equations;
  • Delay differential equations;
  • Fixed point theory;
  • Fractional differential equations;
  • Functional equations;
  • Equations on time scales;
  • Partial differential equations;
  • Fractional differential equations;
  • Stochastic differential equations;
  • Integral equations.

Prof. Dr. Ravi P. Agarwal
Prof. Dr. Maria Alessandra Ragusa
Topic Editors

Participating Journals

Journal Name Impact Factor CiteScore Launched Year First Decision (median) APC
Entropy
entropy
2.7 4.7 1999 20.8 Days CHF 2600 Submit
Fractal and Fractional
fractalfract
5.4 3.6 2017 18.9 Days CHF 2700 Submit
Dynamics
dynamics
- - 2021 12.7 Days CHF 1000 Submit
Mathematics
mathematics
2.4 3.5 2013 16.9 Days CHF 2600 Submit
Computation
computation
2.2 3.3 2013 18 Days CHF 1800 Submit
Axioms
axioms
2.0 - 2012 21.8 Days CHF 2400 Submit

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Published Papers (47 papers)

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13 pages, 395 KiB  
Article
The Soliton Solutions for Nonlocal Multi-Component Higher-Order Gerdjikov–Ivanov Equation via Riemann–Hilbert Problem
by Jinshan Liu, Huanhe Dong, Yong Fang and Yong Zhang
Fractal Fract. 2024, 8(3), 177; https://doi.org/10.3390/fractalfract8030177 - 19 Mar 2024
Viewed by 563
Abstract
The Lax pairs of the higher-order Gerdjikov–Ivanov (HOGI) equation are extended to the multi-component formula. Then, we first derive four different types of nonlocal group reductions to this new system. To construct the solution of these four nonlocal equations, we utilize the Riemann–Hilbert [...] Read more.
The Lax pairs of the higher-order Gerdjikov–Ivanov (HOGI) equation are extended to the multi-component formula. Then, we first derive four different types of nonlocal group reductions to this new system. To construct the solution of these four nonlocal equations, we utilize the Riemann–Hilbert method. Compared to the local HOGI equation, the solutions of nonlocal equations not only depend on the local spatial and time variables, but also the nonlocal variables. To exhibit the dynamic behavior, we consider the reverse-spacetime multi-component HOGI equation and its Riemann–Hilbert problem. When the Riemann–Hilbert problem is regular, the integral form solution can be given. Conversely, the exact solutions can be obtained explicitly. Finally, as concrete examples, the periodic solutions of the two-component nonlocal HOGI equation are given, which is different from the local equation. Full article
(This article belongs to the Topic Advances in Nonlinear Dynamics: Methods and Applications)
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18 pages, 292 KiB  
Article
Global Existence of Chemotaxis-Navier–Stokes System with Logistic Source on the Whole Space R2
by Yuting Xu, Qianfan Liu, Yao Chen, Yang Lei and Minghua Yang
Axioms 2024, 13(3), 171; https://doi.org/10.3390/axioms13030171 - 06 Mar 2024
Viewed by 712
Abstract
In this article, we study the Cauchy problem of the chemotaxis-Navier–Stokes system with the consumption and production of chemosignals with a logistic source. The parameters χ0, ξ0, λ>0 and μ>0. [...] Read more.
In this article, we study the Cauchy problem of the chemotaxis-Navier–Stokes system with the consumption and production of chemosignals with a logistic source. The parameters χ0, ξ0, λ>0 and μ>0. The system is a model that involves double chemosignals; one is an attractant consumed by the cells themselves, and the other is an attractant or a repellent produced by the cells themselves. We prove the global-in-time existence and uniqueness of the weak solution to the system for a large class of initial data on the whole space R2. Full article
(This article belongs to the Topic Advances in Nonlinear Dynamics: Methods and Applications)
7 pages, 242 KiB  
Article
Limit Cycles of Discontinuous Piecewise Differential Hamiltonian Systems Separated by a Straight Line
by Joyce A. Casimiro and Jaume Llibre
Axioms 2024, 13(3), 161; https://doi.org/10.3390/axioms13030161 - 29 Feb 2024
Viewed by 724
Abstract
In this article, we study the maximum number of limit cycles of discontinuous piecewise differential systems, formed by two Hamiltonians systems separated by a straight line. We consider three cases, when both Hamiltonians systems in each side of the discontinuity line have simultaneously [...] Read more.
In this article, we study the maximum number of limit cycles of discontinuous piecewise differential systems, formed by two Hamiltonians systems separated by a straight line. We consider three cases, when both Hamiltonians systems in each side of the discontinuity line have simultaneously degree one, two or three. We obtain that in these three cases, this maximum number is zero, one and three, respectively. Moreover, we prove that there are discontinuous piecewise differential systems realizing these maximum number of limit cycles. Note that we have solved the extension of the 16th Hilbert problem about the maximum number of limit cycles that these three classes of discontinuous piecewise differential systems separated by one straight line and formed by two Hamiltonian systems with a degree either one, two, or three, which such systems can exhibit. Full article
(This article belongs to the Topic Advances in Nonlinear Dynamics: Methods and Applications)
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24 pages, 708 KiB  
Article
Minimal Wave Speed for a Nonlocal Viral Infection Dynamical Model
by Xinzhi Ren, Lili Liu, Tianran Zhang and Xianning Liu
Fractal Fract. 2024, 8(3), 135; https://doi.org/10.3390/fractalfract8030135 - 26 Feb 2024
Viewed by 899
Abstract
To provide insights into the spreading speed and propagation dynamics of viruses within a host, in this paper, we investigate the traveling wave solutions and minimal wave speed for a degenerate viral infection dynamical model with a nonlocal dispersal operator and saturated incidence [...] Read more.
To provide insights into the spreading speed and propagation dynamics of viruses within a host, in this paper, we investigate the traveling wave solutions and minimal wave speed for a degenerate viral infection dynamical model with a nonlocal dispersal operator and saturated incidence rate. It is found that the minimal wave speed c is the threshold that determines the existence of traveling wave solutions. The existence of traveling fronts connecting a virus-free steady state and a positive steady state with wave speed cc is established by using Schauder’s fixed-point theorem, limiting arguments, and the Lyapunov functional. The nonexistence of traveling fronts for c<c is proven by the Laplace transform. In particular, the lower-bound estimation of the traveling wave solutions is provided by adopting a rescaling method and the comparison principle, which is a crucial prerequisite for demonstrating that the traveling semifronts connect to the positive steady state at positive infinity by using the Lyapunov method and is a challenge for some nonlocal models. Moreover, simulations show that the asymptotic spreading speed may be larger than the minimal wave speed and the spread of the virus may be postponed if the diffusion ability or diffusion radius decreases. The spreading speed may be underestimated or overestimated if local dispersal is adopted. Full article
(This article belongs to the Topic Advances in Nonlinear Dynamics: Methods and Applications)
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24 pages, 11988 KiB  
Article
Study on Dynamic Coupling Behavior of End-Meshing Harmonic Reducers
by Tongliang Liu, Jianmin Wen and Yingda Chen
Mathematics 2024, 12(5), 650; https://doi.org/10.3390/math12050650 - 23 Feb 2024
Viewed by 388
Abstract
To study the coupling mechanism and dynamic responses of an end-face-movable gear transmission system under complex excitation, a specific configuration of end-meshing, movable-gear reduction mechanism was used to achieve predetermined rigid-thrust-transmission and mismatched-gear-meshing functions, which solved the inherent defects of traditional harmonic gear [...] Read more.
To study the coupling mechanism and dynamic responses of an end-face-movable gear transmission system under complex excitation, a specific configuration of end-meshing, movable-gear reduction mechanism was used to achieve predetermined rigid-thrust-transmission and mismatched-gear-meshing functions, which solved the inherent defects of traditional harmonic gear mechanisms of thin-wall-flexible wheels that are easily damaged by fatigue. Considering the phenomenon of elastic deformation of live teeth that is accompanied by significant changes in meshing characteristics in the transmission process of an end-meshing harmonic reducer, the influences of dynamic meshing parameters, live tooth deformation, time-varying stiffness of tooth meshing, and time-varying backlash on nonlinear dynamic performance were explored, as well as the mechanisms of multi-parameter coupling effects on transmission performance. The nonlinear dynamics model of the end-meshing harmonic reducer was established to solve the chattering prediction problem. Finally, a comprehensive test bed for the transmission system of a harmonic reducer with a meshing type with an adjustable-characteristic end was built to verify the correctness of the theoretical model and provide the theoretical and technical basis for exploring the optimal parameter selection to address the passive vibration-suppression problem. Full article
(This article belongs to the Topic Advances in Nonlinear Dynamics: Methods and Applications)
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16 pages, 206 KiB  
Article
New Types of Derivative Non-linear Schrödinger Equations Related to Kac–Moody Algebra A2(1)
by Aleksander Aleksiev Stefanov
Dynamics 2024, 4(1), 81-96; https://doi.org/10.3390/dynamics4010005 - 18 Jan 2024
Viewed by 346
Abstract
We derive a new system of integrable derivative non-linear Schrödinger equations with an L operator, quadratic in the spectral parameter with coefficients belonging to the Kac–Moody algebra A2(1). The construction of the fundamental analytic solutions of L is [...] Read more.
We derive a new system of integrable derivative non-linear Schrödinger equations with an L operator, quadratic in the spectral parameter with coefficients belonging to the Kac–Moody algebra A2(1). The construction of the fundamental analytic solutions of L is outlined and they are used to introduce the scattering data, thus formulating the scattering problem for the Lax pair L,M. Full article
(This article belongs to the Topic Advances in Nonlinear Dynamics: Methods and Applications)
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11 pages, 292 KiB  
Article
A ¯-Dressing Method for the Kundu-Nonlinear Schrödinger Equation
by Jiawei Hu and Ning Zhang
Mathematics 2024, 12(2), 278; https://doi.org/10.3390/math12020278 - 15 Jan 2024
Viewed by 484
Abstract
In this paper, we employed the ¯-dressing method to investigate the Kundu-nonlinear Schrödinger equation based on the local 2 × 2 matrix ¯ problem. The Lax spectrum problem is used to derive a singular spectral problem of time and space [...] Read more.
In this paper, we employed the ¯-dressing method to investigate the Kundu-nonlinear Schrödinger equation based on the local 2 × 2 matrix ¯ problem. The Lax spectrum problem is used to derive a singular spectral problem of time and space associated with a Kundu-NLS equation. The N-solitions of the Kundu-NLS equation were obtained based on the ¯ equation by choosing a special spectral transformation matrix, and a gradual analysis of the long-duration behavior of the equation was acquired. Subsequently, the one- and two-soliton solutions of Kundu-NLS equations were obtained explicitly. In optical fiber, due to the wide application of telecommunication and flow control routing systems, people are very interested in the propagation of femtosecond optical pulses, and a high-order, nonlinear Schrödinger equation is needed to build a model. In plasma physics, the soliton equation can predict the modulation instability of light waves in different media. Full article
(This article belongs to the Topic Advances in Nonlinear Dynamics: Methods and Applications)
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31 pages, 8215 KiB  
Article
Exploiting Newly Designed Fractional-Order 3D Lorenz Chaotic System and 2D Discrete Polynomial Hyper-Chaotic Map for High-Performance Multi-Image Encryption
by Wei Feng, Quanwen Wang, Hui Liu, Yu Ren, Junhao Zhang, Shubo Zhang, Kun Qian and Heping Wen
Fractal Fract. 2023, 7(12), 887; https://doi.org/10.3390/fractalfract7120887 - 16 Dec 2023
Cited by 7 | Viewed by 1080
Abstract
Chaos-based image encryption has become a prominent area of research in recent years. In comparison to ordinary chaotic systems, fractional-order chaotic systems tend to have a greater number of control parameters and more complex dynamical characteristics. Thus, an increasing number of researchers are [...] Read more.
Chaos-based image encryption has become a prominent area of research in recent years. In comparison to ordinary chaotic systems, fractional-order chaotic systems tend to have a greater number of control parameters and more complex dynamical characteristics. Thus, an increasing number of researchers are introducing fractional-order chaotic systems to enhance the security of chaos-based image encryption. However, their suggested algorithms still suffer from some security, practicality, and efficiency problems. To address these problems, we first constructed a new fractional-order 3D Lorenz chaotic system and a 2D sinusoidally constrained polynomial hyper-chaotic map (2D-SCPM). Then, we elaborately developed a multi-image encryption algorithm based on the new fractional-order 3D Lorenz chaotic system and 2D-SCPM (MIEA-FCSM). The introduction of the fractional-order 3D Lorenz chaotic system with the fourth parameter not only enables MIEA-FCSM to have a significantly large key space but also enhances its overall security. Compared with recent alternatives, the structure of 2D-SCPM is simpler and more conducive to application implementation. In our proposed MIEA-FCSM, multi-channel fusion initially reduces the number of pixels to one-sixth of the original. Next, after two rounds of plaintext-related chaotic random substitution, dynamic diffusion, and fast scrambling, the fused 2D pixel matrix is eventually encrypted into the ciphertext one. According to numerous experiments and analyses, MIEA-FCSM obtained excellent scores for key space (2541), correlation coefficients (<0.004), information entropy (7.9994), NPCR (99.6098%), and UACI (33.4659%). Significantly, MIEA-FCSM also attained an average encryption rate as high as 168.5608 Mbps. Due to the superiority of the new fractional-order chaotic system, 2D-SCPM, and targeted designs, MIEA-FCSM outperforms many recently reported leading image encryption algorithms. Full article
(This article belongs to the Topic Advances in Nonlinear Dynamics: Methods and Applications)
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23 pages, 326 KiB  
Article
A System of Coupled Impulsive Neutral Functional Differential Equations: New Existence Results Driven by Fractional Brownian Motion and the Wiener Process
by Abdelkader Moumen, Mohamed Ferhat, Amin Benaissa Cherif, Mohamed Bouye and Mohamad Biomy
Mathematics 2023, 11(24), 4949; https://doi.org/10.3390/math11244949 - 13 Dec 2023
Viewed by 607
Abstract
Conditions for the existence and uniqueness of mild solutions for a system of semilinear impulsive differential equations with infinite fractional Brownian movements and the Wiener process are established. Our approach is based on a novel application of Burton and Kirk’s fixed point theorem [...] Read more.
Conditions for the existence and uniqueness of mild solutions for a system of semilinear impulsive differential equations with infinite fractional Brownian movements and the Wiener process are established. Our approach is based on a novel application of Burton and Kirk’s fixed point theorem in extended Banach spaces. This paper aims to extend current results to a differential-inclusions scenario. The motivation of this paper for impulsive neutral differential equations is to investigate the existence of solutions for impulsive neutral differential equations with fractional Brownian motion and a Wiener process (topics that have not been considered and are the main focus of this paper). Full article
(This article belongs to the Topic Advances in Nonlinear Dynamics: Methods and Applications)
16 pages, 8636 KiB  
Article
A Conservative Memristive Chaotic System with Extreme Multistability and Its Application in Image Encryption
by Jian Li, Bo Liang, Xiefu Zhang and Zhixin Yu
Entropy 2023, 25(12), 1656; https://doi.org/10.3390/e25121656 - 13 Dec 2023
Viewed by 803
Abstract
In this work, a novel conservative memristive chaotic system is constructed based on a smooth memristor. In addition to generating multiple types of quasi-periodic trajectories within a small range of a single parameter, the amplitude of the system can be controlled by changing [...] Read more.
In this work, a novel conservative memristive chaotic system is constructed based on a smooth memristor. In addition to generating multiple types of quasi-periodic trajectories within a small range of a single parameter, the amplitude of the system can be controlled by changing the initial values. Moreover, the proposed system exhibits nonlinear dynamic characteristics, involving extreme multistability behavior of isomorphic and isomeric attractors. Finally, the proposed system is implemented using STMicroelectronics 32 and applied to image encryption. The excellent encryption performance of the conservative chaotic system is proven by an average correlation coefficient of 0.0083 and an information entropy of 7.9993, which provides a reference for further research on conservative memristive chaotic systems in the field of image encryption. Full article
(This article belongs to the Topic Advances in Nonlinear Dynamics: Methods and Applications)
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25 pages, 1979 KiB  
Article
Adaptive Backstepping Boundary Control for a Class of Modified Burgers’ Equation
by Francisco Jurado and Oscar F. Murillo-García
Fractal Fract. 2023, 7(12), 834; https://doi.org/10.3390/fractalfract7120834 - 24 Nov 2023
Viewed by 1163
Abstract
Burgers’ equation is used to describe wave phenomena in hydrodynamics and acoustics. It was derived originally as a prototype to provide analytic insight into the nature of turbulence and its modeling, and has found applications in the study of shock waves and wave [...] Read more.
Burgers’ equation is used to describe wave phenomena in hydrodynamics and acoustics. It was derived originally as a prototype to provide analytic insight into the nature of turbulence and its modeling, and has found applications in the study of shock waves and wave transmission. Burgers’ equation is not globally controllable, and under certain conditions it can be neutrally stable. In this study, we explore the adaptive backstepping boundary control (BBC) methodology on a modified Burgers’ equation with unknown parameters, but constant, for the reactive and convective (nonlinear) terms, with Robin and Neumann boundary conditions (BCs), where this latter BC is actuated by the control signal. The nominal controller is designed from a linear partial differential equation (PDE), and under the assumption that this nominal controller also achieves stabilization for the modified Burgers’ equation, then its adaptive version is proposed for the control of such nonlinear PDE systems. Simulation results show convergence near the ideal values for the parametric estimates while the estimation error converges to zero. Full article
(This article belongs to the Topic Advances in Nonlinear Dynamics: Methods and Applications)
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19 pages, 3171 KiB  
Article
Effects of Competition Intensities and R&D Spillovers on a Cournot Duopoly Game of Digital Economies
by Xiaoliang Li, Li Su and Jianjun Wang
Fractal Fract. 2023, 7(10), 737; https://doi.org/10.3390/fractalfract7100737 - 06 Oct 2023
Cited by 1 | Viewed by 806
Abstract
In this paper, we introduce a Cournot duopoly game that can characterize fierce competition in digital economies and employ it to examine the effects of research and development (R&D) spillovers while considering various competition intensities. We obtain the analytical solution of the Nash [...] Read more.
In this paper, we introduce a Cournot duopoly game that can characterize fierce competition in digital economies and employ it to examine the effects of research and development (R&D) spillovers while considering various competition intensities. We obtain the analytical solution of the Nash equilibrium and the expression of commodity price, firm production, and variable profit under some key competition intensities. Furthermore, we analyze the local stability of the Nash equilibrium and derive that the equilibrium may lose its stability only through a 1:4 resonance bifurcation. Numerical simulations are conducted, through which we find that the Nash equilibrium transitions to complex dynamics through a cascade of period-doubling bifurcations. Phase portraits are also provided to illustrate more details of the dynamics, which confirm the previous theoretical finding that the Nash equilibrium loses its stability through a 1:4 resonance bifurcation. Full article
(This article belongs to the Topic Advances in Nonlinear Dynamics: Methods and Applications)
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42 pages, 8062 KiB  
Article
Exploiting the Abstract Calculus Pattern for the Integration of Ordinary Differential Equations for Dynamics Systems: An Object-Oriented Programming Approach in Modern Fortran
by Stefano Zaghi and Cristiano Andolfi
Dynamics 2023, 3(3), 488-529; https://doi.org/10.3390/dynamics3030026 - 28 Aug 2023
Viewed by 567
Abstract
This manuscript relates to the exploiting of the abstract calculus pattern (ACP) for the (numerical) solution of ordinary differential equation (ODEs) systems, which are ubiquitous mathematical formulations of many physical (dynamical) phenomena. We present FOODIE, a software suite aimed to numerically solve ODE [...] Read more.
This manuscript relates to the exploiting of the abstract calculus pattern (ACP) for the (numerical) solution of ordinary differential equation (ODEs) systems, which are ubiquitous mathematical formulations of many physical (dynamical) phenomena. We present FOODIE, a software suite aimed to numerically solve ODE problems by means of a clear, concise, and efficient abstract interface. The results presented prove manifold findings, in particular that our ACP approach enables ease of code development, clearness and robustness, maximization of code re-usability, and conciseness comparable with computer algebra system (CAS) programming (interpreted) but with the computational performance of compiled programming. The proposed programming model is also proven to be agnostic with respect to the parallel paradigm of the computational architecture: the results show that FOODIE applications have good speedup with both shared (OpenMP) and distributed (MPI, CAF) memory architectures. The present paper is the first announcement of the FOODIE project: the current implementation is extensively discussed, and its capabilities are proved by means of tests and examples. Full article
(This article belongs to the Topic Advances in Nonlinear Dynamics: Methods and Applications)
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24 pages, 725 KiB  
Article
Exact Solutions of Nonlinear Second-Order Autonomous Ordinary Differential Equations: Application to Mechanical Systems
by Murillo V. B. Santana
Dynamics 2023, 3(3), 444-467; https://doi.org/10.3390/dynamics3030024 - 14 Aug 2023
Viewed by 1108
Abstract
Many physical processes can be described via nonlinear second-order ordinary differential equations and so, exact solutions to these equations are of interest as, aside from their accuracy, they may reveal beforehand key properties of the system’s response. This work presents a method for [...] Read more.
Many physical processes can be described via nonlinear second-order ordinary differential equations and so, exact solutions to these equations are of interest as, aside from their accuracy, they may reveal beforehand key properties of the system’s response. This work presents a method for computing exact solutions of second-order nonlinear autonomous undamped ordinary differential equations. The solutions are divided into nine cases, each depending on the initial conditions and the system’s first integral. The exact solutions are constructed via a suitable parametrization of the unknown function into a class of functions capable of representing its behavior. The solution is shown to exist and be well-defined in all cases for a general nonlinear form of the differential equation. Practical properties of the solution, such as its period, time to reach an extreme value or long-term behavior, are obtained without the need of computing the solution in advance. Illustrative examples considering different types of nonlinearity present in classical physical systems are used to further validate the obtained exact solutions. Full article
(This article belongs to the Topic Advances in Nonlinear Dynamics: Methods and Applications)
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18 pages, 1038 KiB  
Article
Dynamics of a Fractional-Order COVID-19 Epidemic Model with Quarantine and Standard Incidence Rate
by Trisilowati, Isnani Darti, Raqqasyi Rahmatullah Musafir, Maya Rayungsari and Agus Suryanto
Axioms 2023, 12(6), 591; https://doi.org/10.3390/axioms12060591 - 15 Jun 2023
Cited by 3 | Viewed by 1136
Abstract
In this paper, we propose a fractional-order COVID-19 epidemic model with a quarantine and standard incidence rate using the Caputo fractional-order derivative. The model consists of six classes: susceptible (S), exposed (E), infected (I), quarantined (Q [...] Read more.
In this paper, we propose a fractional-order COVID-19 epidemic model with a quarantine and standard incidence rate using the Caputo fractional-order derivative. The model consists of six classes: susceptible (S), exposed (E), infected (I), quarantined (Q), recovered (R), and deceased (M). In our proposed model, we simultaneously consider the recovery rate and quarantine rate of infected individuals, which has not been considered in other fractional-order COVID-19 epidemic models. Furthermore, we consider the standard incidence rate in the model. For our proposed model, we prove the existence, uniqueness, non-negativity, and boundedness of the solution. The model has two equilibrium points: disease-free equilibrium and endemic equilibrium. Implementing the spectral radius of the next-generation matrix, we obtain the basic reproduction number (R0). The disease-free equilibrium always exists and is locally and globally asymptotically stable only if R0<1. On the other hand, endemic equilibrium exists and is globally asymptotically stable if R0>1. Our numerical simulation confirms the stability properties of the equilibrium. The smaller the order of the derivative, the slower the convergence of the solution of the model. Both the recovery rate and quarantine rate of the infected class are important parameters determining the stability of the equilibrium point. Based on parameter estimation from COVID-19 data in Indonesia, the fractional-order model has better performance than the first-order model for both the calibration and 20-day forecasting of confirmed daily active cases of COVID-19. Full article
(This article belongs to the Topic Advances in Nonlinear Dynamics: Methods and Applications)
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18 pages, 16091 KiB  
Article
Study of Generalized Chaotic Synchronization Method Incorporating Error-Feedback Coefficients
by Yanan Xing, Wenjie Dong, Jian Zeng, Pengteng Guo, Jing Zhang and Qun Ding
Entropy 2023, 25(5), 818; https://doi.org/10.3390/e25050818 - 18 May 2023
Cited by 2 | Viewed by 1026
Abstract
In this paper, taking the generalized synchronization problem of discrete chaotic systems as a starting point, a generalized synchronization method incorporating error-feedback coefficients into the controller based on the generalized chaos synchronization theory and stability theorem for nonlinear systems is proposed. Two discrete [...] Read more.
In this paper, taking the generalized synchronization problem of discrete chaotic systems as a starting point, a generalized synchronization method incorporating error-feedback coefficients into the controller based on the generalized chaos synchronization theory and stability theorem for nonlinear systems is proposed. Two discrete chaotic systems with different dimensions are constructed in this paper, the dynamics of the proposed systems are analyzed, and finally, the phase diagrams, Lyapunov exponent diagrams, and bifurcation diagrams of these are shown and described. The experimental results show that the design of the adaptive generalized synchronization system is achievable in cases in which the error-feedback coefficient satisfies certain conditions. Finally, a chaotic hiding image encryption transmission system based on a generalized synchronization approach is proposed, in which an error-feedback coefficient is introduced into the controller. Full article
(This article belongs to the Topic Advances in Nonlinear Dynamics: Methods and Applications)
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26 pages, 3131 KiB  
Article
Spectral Properties of Exact Polarobreathers in Semiclassical Systems
by Juan F. R. Archilla and Jānis Bajārs
Axioms 2023, 12(5), 437; https://doi.org/10.3390/axioms12050437 - 27 Apr 2023
Cited by 1 | Viewed by 896
Abstract
In this paper, we study the spectral properties of polarobreathers, that is, breathers carrying charge in a one-dimensional semiclassical model. We adapt recently developed numerical methods that preserve the charge probability at every step of time integration without using the Born–Oppenheimer approximation, which [...] Read more.
In this paper, we study the spectral properties of polarobreathers, that is, breathers carrying charge in a one-dimensional semiclassical model. We adapt recently developed numerical methods that preserve the charge probability at every step of time integration without using the Born–Oppenheimer approximation, which is the assumption that the electron is not at equilibrium with the atoms or ions. We develop an algorithm to obtain exact polarobreather solutions. The properties of polarobreathers, both stationary and moving ones, are deduced from the lattice and charge variable spectra in the frequency–momentum space. We consider an efficient approach to produce approximate polarobreathers with long lifespans. Their spectrum allows for the determination of the initial conditions and the necessary parameters to obtain numerically exact polarobreathers. The spectra of exact polarobreathers become extremely simple and easy to interpret. We also solve the problem that the charge frequency is not an observable, but the frequency of the charge probability certainly is an observable. Full article
(This article belongs to the Topic Advances in Nonlinear Dynamics: Methods and Applications)
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17 pages, 900 KiB  
Article
Aperiodic Sampled-Data Control for Anti-Synchronization of Chaotic Nonlinear Systems Subject to Input Saturation
by Meixuan Li and Yingjie Fan
Axioms 2023, 12(4), 403; https://doi.org/10.3390/axioms12040403 - 21 Apr 2023
Cited by 2 | Viewed by 1115
Abstract
This paper studies the aperiodic sampled-data (SD) control anti-synchronization issue of chaotic nonlinear systems under the effects of input saturation. At first, to describe the simultaneous existence of the aperiodic SD pattern and the input saturation, a nonlinear closed-loop system model is established. [...] Read more.
This paper studies the aperiodic sampled-data (SD) control anti-synchronization issue of chaotic nonlinear systems under the effects of input saturation. At first, to describe the simultaneous existence of the aperiodic SD pattern and the input saturation, a nonlinear closed-loop system model is established. Then, to make the anti-synchronization analysis, a relaxed sampling-interval-dependent Lyapunov functional (RSIDLF) is constructed for the resulting closed-loop system. Thereinto, the positive definiteness requirement of the RSIDLF is abandoned. Due to the indefiniteness of RSIDLF, the discrete-time Lyapunov method (DTLM) then is used to guarantee the local stability of the trivial solutions of the modeled nonlinear system. Furthermore, two convex optimization schemes are proposed to expand the allowable initial area (AIA) and maximize the upper bound of the sampling period (UBSP). Finally, two examples of nonlinear systems are provided to illustrate the superiority of the RSIDLF method over the previous methods in expanding the AIA and enlarging the UBSP. Full article
(This article belongs to the Topic Advances in Nonlinear Dynamics: Methods and Applications)
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19 pages, 27755 KiB  
Article
Fear Effect on a Predator–Prey Model with Non-Differential Fractional Functional Response
by Salam Mohammed Ghazi Al-Mohanna and Yong-Hui Xia
Fractal Fract. 2023, 7(4), 312; https://doi.org/10.3390/fractalfract7040312 - 04 Apr 2023
Viewed by 1285
Abstract
In this paper, we study the factor of the fear effect in a predator–prey model with prey refuge and a non-differentiable fractional functional response due to the group defense. Since the functional response is non-differentiable, the dynamics of this system are considerably different [...] Read more.
In this paper, we study the factor of the fear effect in a predator–prey model with prey refuge and a non-differentiable fractional functional response due to the group defense. Since the functional response is non-differentiable, the dynamics of this system are considerably different from the dynamics of a classical predator–prey system. The persistence, the stability and the existence of the steady states are investigated. We examine the Hopf bifurcation at the unique positive equilibrium. Direct Hopf bifurcation is studied via the central manifold theorem. When the value of the fear factor decreases and is less than a threshold κH, the limit cycle appears, and it disappears through a loop of heteroclinic orbits when the value of the fear factor is equal to a value κhet. Full article
(This article belongs to the Topic Advances in Nonlinear Dynamics: Methods and Applications)
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31 pages, 511 KiB  
Article
Moderate Averaged Deviations for a Multi-Scale System with Jumps and Memory
by André de Oliveira Gomes and Pedro Catuogno
Dynamics 2023, 3(1), 171-201; https://doi.org/10.3390/dynamics3010011 - 14 Mar 2023
Cited by 1 | Viewed by 1321
Abstract
This work studies a two-time-scale functional system given by two jump diffusions under the scale separation by a small parameter ε0. The coefficients of the equations that govern the dynamics of the system depend on the segment process of the [...] Read more.
This work studies a two-time-scale functional system given by two jump diffusions under the scale separation by a small parameter ε0. The coefficients of the equations that govern the dynamics of the system depend on the segment process of the slow variable (responsible for capturing delay effects on the slow component) and on the state of the fast variable. We derive a moderate deviation principle for the slow component of the system in the small noise limit using the weak convergence approach. The rate function is written in terms of the averaged dynamics associated with the multi-scale system. The core of the proof of the moderate deviation principle is the establishment of an averaging principle for the auxiliary controlled processes associated with the slow variable in the framework of the weak convergence approach. The controlled version of the averaging principle for the jump multi-scale diffusion relies on a discretization method inspired by the classical Khasminkii’s averaging principle. Full article
(This article belongs to the Topic Advances in Nonlinear Dynamics: Methods and Applications)
21 pages, 5549 KiB  
Article
Fractional-Step Method with Interpolation for Solving a System of First-Order 2D Hyperbolic Delay Differential Equations
by Karthick Sampath, Subburayan Veerasamy and Ravi P. Agarwal
Computation 2023, 11(3), 57; https://doi.org/10.3390/computation11030057 - 09 Mar 2023
Cited by 1 | Viewed by 1493
Abstract
In this article, we consider a delayed system of first-order hyperbolic differential equations. The presence of the delay term in first-order hyperbolic delay differential equations poses significant challenges in both analysis and numerical solutions. The delay term also makes it more difficult to [...] Read more.
In this article, we consider a delayed system of first-order hyperbolic differential equations. The presence of the delay term in first-order hyperbolic delay differential equations poses significant challenges in both analysis and numerical solutions. The delay term also makes it more difficult to use standard numerical methods for solving differential equations, as these methods often require that the differential equation be evaluated at the current time step. To overcome these challenges, specialized numerical methods and analytical techniques have been developed for solving first-order hyperbolic delay differential equations. We investigated and presented analytical results, such as the maximum principle and stability results. The propagation of discontinuities in the solution was also discussed, providing a framework for understanding its behavior. We presented a fractional-step method using a backward finite difference scheme and showed that the scheme is almost first-order convergent in space and time through the derivation of the error estimate. Additionally, we demonstrated an application of the proposed method to the problem of variable delay differential equations. We demonstrated the practical application of the proposed method to solving variable delay differential equations. The proposed algorithm is based on a numerical approximation method that utilizes a finite difference scheme to discretize the differential equation. We validated our theoretical results through numerical experiments. Full article
(This article belongs to the Topic Advances in Nonlinear Dynamics: Methods and Applications)
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20 pages, 397 KiB  
Article
Finite-Time H Control for Time-Delay Markovian Jump Systems with Partially Unknown Transition Rate via General Controllers
by Xikui Liu, Xinye Guo, Wencheng Liu and Yan Li
Entropy 2023, 25(3), 402; https://doi.org/10.3390/e25030402 - 22 Feb 2023
Viewed by 1191
Abstract
This paper deals with the problems of finite-time boundedness (FTB) and H FTB for time-delay Markovian jump systems with a partially unknown transition rate. First of all, sufficient conditions are provided, ensuring the FTB and H FTB of systems given by [...] Read more.
This paper deals with the problems of finite-time boundedness (FTB) and H FTB for time-delay Markovian jump systems with a partially unknown transition rate. First of all, sufficient conditions are provided, ensuring the FTB and H FTB of systems given by linear matrix inequalities (LMIs). A new type of partially delay-dependent controller (PDDC) is designed so that the resulting closed-loop systems are finite-time bounded and satisfy a given H disturbance attenuation level. The PDDC contains both non-time-delay and time-delay states, though not happening at the same time, which is related to the probability distribution of the Bernoulli variable. Furthermore, the PDDC is extended to two other cases; one does not contain the Bernoulli variable, and the other experiences a disordering phenomenon. Finally, three numerical examples are used to show the effectiveness of the proposed approaches. Full article
(This article belongs to the Topic Advances in Nonlinear Dynamics: Methods and Applications)
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20 pages, 8850 KiB  
Article
Implementation of the Simple Hyperchaotic Memristor Circuit with Attractor Evolution and Large-Scale Parameter Permission
by Gang Yang, Xiaohong Zhang and Ata Jahangir Moshayedi
Entropy 2023, 25(2), 203; https://doi.org/10.3390/e25020203 - 19 Jan 2023
Cited by 6 | Viewed by 1563
Abstract
A novel, simple, four-dimensional hyperchaotic memristor circuit consisting of two capacitors, an inductor and a magnetically controlled memristor is designed. Three parameters (a, b, c) are especially set as the research objects of the model through numerical simulation. It [...] Read more.
A novel, simple, four-dimensional hyperchaotic memristor circuit consisting of two capacitors, an inductor and a magnetically controlled memristor is designed. Three parameters (a, b, c) are especially set as the research objects of the model through numerical simulation. It is found that the circuit not only exhibits a rich attractor evolution phenomenon, but also has large-scale parameter permission. At the same time, the spectral entropy complexity of the circuit is analyzed, and it is confirmed that the circuit contains a significant amount of dynamical behavior. By setting the internal parameters of the circuit to remain constant, a number of coexisting attractors are found under symmetric initial conditions. Then, the results of the attractor basin further confirm the coexisting attractor behavior and multiple stability. Finally, the simple memristor chaotic circuit is designed by the time-domain method with FPGA technology and the experimental results have the same phase trajectory as the numerical calculation results. Hyperchaos and broad parameter selection mean that the simple memristor model has more complex dynamic behavior, which can be widely used in the future, in areas such as secure communication, intelligent control and memory storage. Full article
(This article belongs to the Topic Advances in Nonlinear Dynamics: Methods and Applications)
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22 pages, 7194 KiB  
Article
Saturated Nonsingular Fast Sliding Mode Control for the Crane-Form Pipeline System
by Baigeng Wang and Shurong Li
Entropy 2022, 24(12), 1800; https://doi.org/10.3390/e24121800 - 09 Dec 2022
Viewed by 988
Abstract
The crane-form pipeline (CFP) system is a kind of petrochemical mechanical equipment composed of multiple rotating joints and rigid pipelines. It is often used to transport chemical fluid products in the factory to tank trucks. In order to realize the automatic alignment of [...] Read more.
The crane-form pipeline (CFP) system is a kind of petrochemical mechanical equipment composed of multiple rotating joints and rigid pipelines. It is often used to transport chemical fluid products in the factory to tank trucks. In order to realize the automatic alignment of the CFP and the tank mouth, the trajectory tracking control problem of the CFP must be solved. Therefore, a saturated nonsingular fast terminal sliding mode (SNFTSM) algorithm is proposed in this paper. The new sliding mode manifold is constructed by the nonsingular fast terminal sliding mode (NFTSM) manifold, saturation functions and signum functions. Further, according to the sliding mode control algorithm and the dynamic model of the CFP system, the SNFTSM controller is designed. Owing to the existence of saturation functions in the controller, the stability analysis using the Lyapunov equation needs to be discussed in different cases. The results show that the system states can converge to the equilibrium point in finite time no matter where they are on the state’s phase plane. However, due to the existence of signum functions, the control signal will produce chattering. In order to eliminate the chattering problem, the form of the controller is improved by using the boundary layer function. Finally, the control effect of the algorithm is verified by simulation and compared with the NTSM, NFTSM and SNTSM algorithms. From the comparison results, it is obvious that the controller based on the SNFTSM algorithm can effectively reduce the amplitude of the control torque while guaranteeing the fast convergence of the CFP system state error. Specifically, compared with the NFTSM algorithm, the maximum input torque can even be reduced by more than half. Full article
(This article belongs to the Topic Advances in Nonlinear Dynamics: Methods and Applications)
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11 pages, 330 KiB  
Article
An Extension on the Local Convergence for the Multi-Step Seventh Order Method with ψ-Continuity Condition in the Banach Spaces
by Mohammad Taghi Darvishi, R. H. Al-Obaidi, Akanksha Saxena, Jai Prakash Jaiswal and Kamal Raj Pardasani
Fractal Fract. 2022, 6(12), 713; https://doi.org/10.3390/fractalfract6120713 - 30 Nov 2022
Viewed by 1072
Abstract
The local convergence analysis of the multi-step seventh order method to solve nonlinear equations is presented in this paper. The point of this paper is that our proposed study requires a weak hypothesis where the Fréchet derivative of the nonlinear operator satisfies the [...] Read more.
The local convergence analysis of the multi-step seventh order method to solve nonlinear equations is presented in this paper. The point of this paper is that our proposed study requires a weak hypothesis where the Fréchet derivative of the nonlinear operator satisfies the ψ-continuity condition, which thereby extends the applicability of the method when both Lipschitz and Hölder conditions fail. The convergence in this study is considered under the hypotheses on the first-order derivative without involving derivatives of the higher-order. To find a subset of the original convergence domain, a strategy is devised here. As a result, the new Lipschitz constants are at least as tight as the old ones, allowing for a more precise convergence analysis in the local convergence case. Some concrete numerical examples showing the performance of the method over some existing schemes are presented in this article. Full article
(This article belongs to the Topic Advances in Nonlinear Dynamics: Methods and Applications)
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20 pages, 3271 KiB  
Article
Distinguish between Stochastic and Chaotic Signals by a Local Structure-Based Entropy
by Zelin Zhang, Jun Wu, Yufeng Chen, Ji Wang and Jinyu Xu
Entropy 2022, 24(12), 1752; https://doi.org/10.3390/e24121752 - 30 Nov 2022
Viewed by 1521
Abstract
As a measure of complexity, information entropy is frequently used to categorize time series, such as machinery failure diagnostics, biological signal identification, etc., and is thought of as a characteristic of dynamic systems. Many entropies, however, are ineffective for multivariate scenarios due to [...] Read more.
As a measure of complexity, information entropy is frequently used to categorize time series, such as machinery failure diagnostics, biological signal identification, etc., and is thought of as a characteristic of dynamic systems. Many entropies, however, are ineffective for multivariate scenarios due to correlations. In this paper, we propose a local structure entropy (LSE) based on the idea of a recurrence network. Given certain tolerance and scales, LSE values can distinguish multivariate chaotic sequences between stochastic signals. Three financial market indices are used to evaluate the proposed LSE. The results show that the LSEFSTE100 and LSES&P500 are higher than LSESZI, which indicates that the European and American stock markets are more sophisticated than the Chinese stock market. Additionally, using decision trees as the classifiers, LSE is employed to detect bearing faults. LSE performs higher on recognition accuracy when compared to permutation entropy. Full article
(This article belongs to the Topic Advances in Nonlinear Dynamics: Methods and Applications)
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20 pages, 3245 KiB  
Article
Adaptive Dynamic Surface Control of Strict-Feedback Fractional-Order Nonlinear Systems with Input Quantization and External Disturbances
by Fan Zhang, Xiongfeng Deng and Lisheng Wei
Fractal Fract. 2022, 6(12), 698; https://doi.org/10.3390/fractalfract6120698 - 25 Nov 2022
Cited by 1 | Viewed by 1080
Abstract
In this work, an adaptive dynamic surface control law for a type of strict-feedback fractional-order nonlinear system is proposed. The considered system contained input quantization and unknown external disturbances. The virtual control law is presented by utilizing a dynamic surface control approach at [...] Read more.
In this work, an adaptive dynamic surface control law for a type of strict-feedback fractional-order nonlinear system is proposed. The considered system contained input quantization and unknown external disturbances. The virtual control law is presented by utilizing a dynamic surface control approach at each step, where the nonlinear compensating term with the estimation of unknown bounded parameters is introduced to overcome the influence of unknown external disturbances and surface errors. Meanwhile, the adaptive laws of relevant parameters are also designed. In addition, an improved fractional-order nonlinear filter is developed to deal with the explosion of complexity raised by the recursive process. In the last step, an adaptive dynamic surface control law is proposed to ensure the convergence of tracking error, in which the Nussbaum gain function is applied to solve the problem of the unknown control gain generated by input quantization. Then, the fractional Lyapunov stability theory is applied to verify the stability of the proposed control law. Finally, simulation examples are given to illustrate the effectiveness of the proposed control law. Full article
(This article belongs to the Topic Advances in Nonlinear Dynamics: Methods and Applications)
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18 pages, 1202 KiB  
Article
A Malware Propagation Model Considering Conformity Psychology in Social Networks
by Qingyi Zhu, Yuhang Liu, Xuhang Luo and Kefei Cheng
Axioms 2022, 11(11), 632; https://doi.org/10.3390/axioms11110632 - 10 Nov 2022
Cited by 1 | Viewed by 1680
Abstract
At present, malware is still a major security threat to computer networks. However, only a fraction of users with some security consciousness take security measures to protect computers on their own initiative, and others who know the current situation through social networks usually [...] Read more.
At present, malware is still a major security threat to computer networks. However, only a fraction of users with some security consciousness take security measures to protect computers on their own initiative, and others who know the current situation through social networks usually follow suit. This phenomenon is referred to as conformity psychology. It is obvious that more users will take countermeasures to prevent computers from being infected if the malware spreads to a certain extent. This paper proposes a deterministic nonlinear SEIQR propagation model to investigate the impact of conformity psychology on malware propagation. Both the local and global stabilities of malware-free equilibrium are proven while the existence and local stability of endemic equilibrium is proven by using the central manifold theory. Additionally, some numerical examples and simulation experiments based on two network datasets are performed to verify the theoretical analysis results. Finally, the sensitivity analysis of system parameters is carried out. Full article
(This article belongs to the Topic Advances in Nonlinear Dynamics: Methods and Applications)
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17 pages, 800 KiB  
Article
Dynamical Behaviors of an SIR Epidemic Model with Discrete Time
by Bo Li, Zohreh Eskandari and Zakieh Avazzadeh
Fractal Fract. 2022, 6(11), 659; https://doi.org/10.3390/fractalfract6110659 - 07 Nov 2022
Cited by 31 | Viewed by 2124
Abstract
Analytically and numerically, the study examines the stability and local bifurcations of a discrete-time SIR epidemic model. For this model, a number of bifurcations are studied, including the transcritical, flip bifurcations, Neimark–Sacker bifurcations, and strong resonances. These bifurcations are checked, and [...] Read more.
Analytically and numerically, the study examines the stability and local bifurcations of a discrete-time SIR epidemic model. For this model, a number of bifurcations are studied, including the transcritical, flip bifurcations, Neimark–Sacker bifurcations, and strong resonances. These bifurcations are checked, and their non-degeneracy conditions are determined by using the normal form technique (computing of critical normal form coefficients). We use the MATLAB toolbox MatcontM, which is based on the numerical continuation method, to confirm the obtained analytical results and specify more complex behaviors of the model. Numerical simulation is employed to present a closed invariant curve emerging from a Neimark–Sacker point and its breaking down to several closed invariant curves and eventually giving rise to a chaotic strange attractor by increasing the bifurcation parameter. Full article
(This article belongs to the Topic Advances in Nonlinear Dynamics: Methods and Applications)
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18 pages, 14064 KiB  
Article
Stability Analysis of Fractional-Order Mathieu Equation with Forced Excitation
by Ruihong Mu, Shaofang Wen, Yongjun Shen and Chundi Si
Fractal Fract. 2022, 6(11), 633; https://doi.org/10.3390/fractalfract6110633 - 31 Oct 2022
Cited by 1 | Viewed by 1451
Abstract
The advantage of fractional-order derivative has attracted extensive attention in the field of dynamics. In this paper, we investigated the stability of the fractional-order Mathieu equation under forced excitation, which is based on a model of the pantograph–catenary system. First, we obtained the [...] Read more.
The advantage of fractional-order derivative has attracted extensive attention in the field of dynamics. In this paper, we investigated the stability of the fractional-order Mathieu equation under forced excitation, which is based on a model of the pantograph–catenary system. First, we obtained the approximate analytical expressions and periodic solutions of the stability boundaries by the multi-scale method and the perturbation method, and the correctness of these results were verified through numerical analysis by Matlab. In addition, by analyzing the stability of the k’T-periodic solutions in the system, we verified the existence of the unstable k’T-resonance lines through numerical simulation, and visually investigated the effect of the system parameters. The results show that forced excitation with a finite period does not change the position of the stability boundaries, but it can affect the expressions of the periodic solutions. Moreover, by analyzing the properties of the resonant lines, we found that when the points with k’T-periodic solutions were perturbed by the same frequency of forced excitation, these points became unstable due to resonance. Finally, we found that both the damping coefficient and the fractional-order parameters in the system have important influences on the stability boundaries and the resonance lines. Full article
(This article belongs to the Topic Advances in Nonlinear Dynamics: Methods and Applications)
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10 pages, 709 KiB  
Article
Infinite Turing Bifurcations in Chains of Van der Pol Systems
by Sergey Kashchenko
Mathematics 2022, 10(20), 3769; https://doi.org/10.3390/math10203769 - 13 Oct 2022
Cited by 3 | Viewed by 1036
Abstract
A chain of coupled systems of Van der Pol equations is considered. We study the local dynamics of this chain in the vicinity of the zero equilibrium state. We make a transition to the system with a continuous spatial variable assuming that the [...] Read more.
A chain of coupled systems of Van der Pol equations is considered. We study the local dynamics of this chain in the vicinity of the zero equilibrium state. We make a transition to the system with a continuous spatial variable assuming that the number of elements in the chain is large enough. The critical cases corresponding to the Turing bifurcations are identified. It is shown that they have infinite dimension. Special nonlinear parabolic equations are proposed on the basis of the asymptotic algorithm. Their nonlocal dynamics describes the local behavior of solutions to the original system. In a number of cases, normalized parabolic equations with two spatial variables arise while considering the most important diffusion type couplings. It has been established, for example, that for the considered systems with a large number of elements, the dynamics change significantly with a slight change in the number of such elements. Full article
(This article belongs to the Topic Advances in Nonlinear Dynamics: Methods and Applications)
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15 pages, 1199 KiB  
Article
Multi-Sensor Scheduling Method Based on Joint Risk Assessment with Variable Weight
by Lin Zhou, Jiawei Wu, Qian Wei, Wentao Shi and Yong Jin
Entropy 2022, 24(9), 1315; https://doi.org/10.3390/e24091315 - 19 Sep 2022
Viewed by 1458
Abstract
In multi-sensor cooperative detection systems, to reduce target threat risk caused by attack tasks and target loss risk induced by uncertain environmental factors, this paper proposes a multi-sensor scheduling method based on joint risk assessment with variable weight. Firstly, considering the target state [...] Read more.
In multi-sensor cooperative detection systems, to reduce target threat risk caused by attack tasks and target loss risk induced by uncertain environmental factors, this paper proposes a multi-sensor scheduling method based on joint risk assessment with variable weight. Firstly, considering the target state and prior expert experience of sensor scheduling, this paper gives a new scheme of target threat risk. Then, by combining the given target threat risk and the target loss risk, this paper constructs a joint risk model to meet the diversity of risk assessment. Secondly, a variable-weighted joint risk assessment model is given based on the adaptive weight of target loss risk and target threat risk, and the optimization problem of multi-sensor scheduling is described to minimize the multi-step prediction of the variable-weighted joint risk model. Finally, this paper relaxes above the non-convex optimization problem as a subconvex problem and designs the scheme of multi-sensor scheduling, improving the rapidity and optimization of the sensor scheduling solution. The simulation results show that the proposed method can adaptively schedule sensors and accurately track targets by using minimum sensor resources. Full article
(This article belongs to the Topic Advances in Nonlinear Dynamics: Methods and Applications)
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29 pages, 8435 KiB  
Article
Rational and Semi-Rational Solutions to the (2 + 1)-Dimensional Maccari System
by Yong Zhang, Huan-He Dong and Yong Fang
Axioms 2022, 11(9), 472; https://doi.org/10.3390/axioms11090472 - 15 Sep 2022
Cited by 1 | Viewed by 1530
Abstract
The KP hierarchy reduction method is one of the most reliable and efficient techniques for determining exact solitary wave solutions to nonlinear partial differential equations. In this paper, according to the KP hierarchy reduction technique, rational and some other semi-rational solutions to the [...] Read more.
The KP hierarchy reduction method is one of the most reliable and efficient techniques for determining exact solitary wave solutions to nonlinear partial differential equations. In this paper, according to the KP hierarchy reduction technique, rational and some other semi-rational solutions to the (2 + 1)-dimensional Maccari system are investigated. It is shown that two different types of breathers can be derived, and under appropriate parameter constraints, they can be reduced to some well known solutions, involving the homoclinic orbits, dark soliton or anti-dark soliton solution. For the dark and anti-dark solution, its interaction is similar to a resonance soliton. Furthermore, by using a limiting technique, we derive two kinds of rational solutions, one is the lump and the other one is the rogue wave. After constructing these solutions, we further discuss the interactions between the obtained solutions. It is interesting that we obtain a parallel breather and a intersectional breather, which seems very surprising. Finally, we also provide a new three-state interaction, which is composed by the dark-soliton, rogue wave and breather and has never been provided for the Maccari system. Full article
(This article belongs to the Topic Advances in Nonlinear Dynamics: Methods and Applications)
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10 pages, 1130 KiB  
Article
Dynamics and Entropy Analysis of a Frictionally Loaded Pendulum
by Grzegorz Litak, Marek Borowiec and Krzysztof Da̧bek
Entropy 2022, 24(9), 1269; https://doi.org/10.3390/e24091269 - 09 Sep 2022
Cited by 1 | Viewed by 1415
Abstract
We use friction to simultaneously damp and excite a pendulum system. A Froude pendulum attached to a suspension shaft is subjected to a frictional load. We investigate two types of response of the system: regular and chaotic responses, depending on the excitation frequency. [...] Read more.
We use friction to simultaneously damp and excite a pendulum system. A Froude pendulum attached to a suspension shaft is subjected to a frictional load. We investigate two types of response of the system: regular and chaotic responses, depending on the excitation frequency. A transient chaotic solution was also obtained. We identify the motions using phase portraits, Poincaré maps, and Fourier spectra. Finally, the composite multiscaled entropy was estimated for the specified cases to confirm the preliminary classification. Full article
(This article belongs to the Topic Advances in Nonlinear Dynamics: Methods and Applications)
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22 pages, 4670 KiB  
Article
Adaptive Neural Network Finite-Time Control of Uncertain Fractional-Order Systems with Unknown Dead-Zone Fault via Command Filter
by Xiongfeng Deng and Lisheng Wei
Fractal Fract. 2022, 6(9), 494; https://doi.org/10.3390/fractalfract6090494 - 04 Sep 2022
Cited by 3 | Viewed by 1447
Abstract
In this paper, the adaptive finite-time control problem for fractional-order systems with uncertainties and unknown dead-zone fault was studied by combining a fractional-order command filter, radial basis function neural network, and Nussbaum gain function technique. First, the fractional-order command filter-based backstepping control method [...] Read more.
In this paper, the adaptive finite-time control problem for fractional-order systems with uncertainties and unknown dead-zone fault was studied by combining a fractional-order command filter, radial basis function neural network, and Nussbaum gain function technique. First, the fractional-order command filter-based backstepping control method is applied to avoid the computational complexity problem existing in the conventional recursive procedure, where the fractional-order command filter is introduced to obtain the filter signals and their fractional-order derivatives. Second, the radial basis function neural network is used to handle the uncertain nonlinear functions in the recursive design step. Third, the Nussbaum gain function technique is considered to handle the unknown control gain caused by the unknown dead-zone fault. Moreover, by introducing the compensating signal into the control law design, the virtual control law, adaptive laws, and the adaptive neural network finite-time control law are constructed to ensure that all signals associated with the closed-loop system are bounded in finite time and that the tracking error can converge to a small neighborhood of origin in finite time. Finally, the validity of the proposed control law is confirmed by providing simulation cases. Full article
(This article belongs to the Topic Advances in Nonlinear Dynamics: Methods and Applications)
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25 pages, 659 KiB  
Article
Event-Triggered Non-PDC Filter Design of Fuzzy Markovian Jump Systems under Mismatch Phenomena
by Khanh Hieu Nguyen and Sung Hyun Kim
Mathematics 2022, 10(16), 2917; https://doi.org/10.3390/math10162917 - 13 Aug 2022
Cited by 3 | Viewed by 1322
Abstract
This paper focuses on dealing with the problem of co-designing a fuzzy-basis-dependent event generator and an asynchronous filter of fuzzy Markovian jump systems via event-triggered non-parallel distribution compensation (non-PDC) scheme. The introduction of the event-triggered non-PDC scheme can reduce the number of real-time [...] Read more.
This paper focuses on dealing with the problem of co-designing a fuzzy-basis-dependent event generator and an asynchronous filter of fuzzy Markovian jump systems via event-triggered non-parallel distribution compensation (non-PDC) scheme. The introduction of the event-triggered non-PDC scheme can reduce the number of real-time filter gain design operations with a large computational load. Furthermore, to perform an effective relaxation process, several kinds of time-varying parameters in filter design conditions are simultaneously relaxed by utilizing two zero equalities of transition probabilities and mismatch errors. In addition, to improve the considered performance, the event generation function is established based on fuzzy-basis-dependent event weighting matrices. Full article
(This article belongs to the Topic Advances in Nonlinear Dynamics: Methods and Applications)
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22 pages, 1147 KiB  
Article
Constrained Optimal Control for Nonlinear Multi-Input Safety-Critical Systems with Time-Varying Safety Constraints
by Jinguang Wang, Chunbin Qin, Xiaopeng Qiao, Dehua Zhang, Zhongwei Zhang, Ziyang Shang and Heyang Zhu
Mathematics 2022, 10(15), 2744; https://doi.org/10.3390/math10152744 - 03 Aug 2022
Cited by 1 | Viewed by 1620
Abstract
In this paper, we investigate the constrained optimal control problem of nonlinear multi-input safety-critical systems with uncertain disturbances and time-varying safety constraints. By utilizing a barrier function transformation, together with a new disturbance-related term and a smooth safety boundary function, a nominal system-dependent [...] Read more.
In this paper, we investigate the constrained optimal control problem of nonlinear multi-input safety-critical systems with uncertain disturbances and time-varying safety constraints. By utilizing a barrier function transformation, together with a new disturbance-related term and a smooth safety boundary function, a nominal system-dependent multi-input barrier transformation architecture is developed to deal with the time-varying safety constraints and uncertain disturbances. Based on the obtained transformation system, the coupled Hamilton–Jacobi–Bellman (HJB) function is established to obtain the constrained Nash equilibrium solution. In addition, due to the fact that it is difficult to solve the HJB function directly, the single critic neural network (NN) is constructed to approximate the optimal performance index function of different control inputs, respectively. It is proved theoretically that, under the influence of uncertain disturbances and time-varying safety constraints, the system states and neural network parameters can be uniformly ultimately bounded (UUB) by the proposed neural network approximation method. Finally, the effectiveness of the proposed method is verified by two nonlinear simulation examples. Full article
(This article belongs to the Topic Advances in Nonlinear Dynamics: Methods and Applications)
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12 pages, 1099 KiB  
Article
Collective Behaviors of Star-Coupled Harmonic Oscillators with Fluctuating Frequency in the Presence of Stochastic Resonance
by Ruibin Ren and George X. Yuan
Fractal Fract. 2022, 6(8), 414; https://doi.org/10.3390/fractalfract6080414 - 28 Jul 2022
Cited by 2 | Viewed by 1333
Abstract
The stochastic resonance (SR) of a star-coupled harmonic oscillator subject to multiplicative fluctuation and periodic force in viscous media is studied. The multiplicative noise is modeled as a dichotomous noise and the memory of viscous media is characterized by a fractional power kernel [...] Read more.
The stochastic resonance (SR) of a star-coupled harmonic oscillator subject to multiplicative fluctuation and periodic force in viscous media is studied. The multiplicative noise is modeled as a dichotomous noise and the memory of viscous media is characterized by a fractional power kernel function. By using the Shapiro–Loginov formula and Laplace transform, we obtain the analytical expressions of the first moment of the steady-state response and study the relationship between the system response and the system parameters in the long-time limit. The simulation results show the nonmonotonic dependence between the response output gain and the input signal frequency, the noise parameters of the system, etc., which indicates that the bona fide resonance and the generalized SR phenomena appear. Furthermore, the fluctuation noise, the number of particles, and the fractional order work together, producing more complex dynamic phenomena compared with the integral-order system. In addition, all the theoretical analyses are supported by the corresponding numerical simulations. We believe that the results that we have found may be a certain reference value for the research and development of the SR. Full article
(This article belongs to the Topic Advances in Nonlinear Dynamics: Methods and Applications)
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28 pages, 7870 KiB  
Article
An Encryption Algorithm for Region of Interest in Medical DICOM Based on One-Dimensional eλ-cos-cot Map
by Xin Meng, Jinqing Li, Xiaoqiang Di, Yaohui Sheng and Donghua Jiang
Entropy 2022, 24(7), 901; https://doi.org/10.3390/e24070901 - 29 Jun 2022
Cited by 6 | Viewed by 1924
Abstract
Today, with the rapid development of the Internet, improving image security becomes more and more important. To improve image encryption efficiency, a novel region of interest (ROI) encryption algorithm based on a chaotic system was proposed. First, a new 1D eλ-cos-cot [...] Read more.
Today, with the rapid development of the Internet, improving image security becomes more and more important. To improve image encryption efficiency, a novel region of interest (ROI) encryption algorithm based on a chaotic system was proposed. First, a new 1D eλ-cos-cot (1D-ECC) with better chaotic performance than the traditional chaotic system is proposed. Second, the chaotic system is used to generate a plaintext-relate keystream based on the label information of a medical image DICOM (Digital Imaging and Communications in Medicine) file, the medical image is segmented using an adaptive threshold, and the segmented region of interest is encrypted. The encryption process is divided into two stages: scrambling and diffusion. In the scrambling stage, helical scanning and index scrambling are combined to scramble. In the diffusion stage, two-dimensional bi-directional diffusion is adopted, that is, the image is bi-directionally diffused row by column to make image security better. The algorithm offers good encryption speed and security performance, according to simulation results and security analysis. Full article
(This article belongs to the Topic Advances in Nonlinear Dynamics: Methods and Applications)
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22 pages, 759 KiB  
Article
The Multicomponent Higher-Order Chen–Lee–Liu System: The Riemann–Hilbert Problem and Its N-Soliton Solution
by Yong Zhang, Huanhe Dong and Yong Fang
Fractal Fract. 2022, 6(6), 327; https://doi.org/10.3390/fractalfract6060327 - 13 Jun 2022
Cited by 1 | Viewed by 1666
Abstract
It is well known that multicomponent integrable systems provide a method for analyzing phenomena with numerous interactions, due to the interactions between their different components. In this paper, we derive the multicomponent higher-order Chen–Lee–Liu (mHOCLL) system through the zero-curvature equation and recursive operators. [...] Read more.
It is well known that multicomponent integrable systems provide a method for analyzing phenomena with numerous interactions, due to the interactions between their different components. In this paper, we derive the multicomponent higher-order Chen–Lee–Liu (mHOCLL) system through the zero-curvature equation and recursive operators. Then, we apply the trace identity to obtain the bi-Hamiltonian structure of mHOCLL system, which certifies that the constructed system is integrable. Considering the spectral problem of the Lax pair, a related Riemann–Hilbert (RH) problem of this integrable system is naturally constructed with zero background, and the symmetry of this spectral problem is given. On the one hand, the explicit expression for the mHOCLL solution is not available when the RH problem is regular. However, according to the formal solution obtained using the Plemelj formula, the long-time asymptotic state of the mHOCLL solution can be obtained. On the other hand, the N-soliton solutions can be explicitly gained when the scattering problem is reflectionless, and its long-time behavior can still be discussed. Finally, the determinant form of the N-soliton solution is given, and one-, two-, and three-soliton solutions as specific examples are shown via the figures. Full article
(This article belongs to the Topic Advances in Nonlinear Dynamics: Methods and Applications)
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13 pages, 959 KiB  
Article
Maximum Power Point Tracking Control for Non-Gaussian Wind Energy Conversion System by Using Survival Information Potential
by Liping Yin, Lanlan Lai, Zhengju Zhu and Tao Li
Entropy 2022, 24(6), 818; https://doi.org/10.3390/e24060818 - 11 Jun 2022
Cited by 4 | Viewed by 1723
Abstract
In this paper, a wind energy conversion system is studied to improve the conversion efficiency and maximize power output. Firstly, a nonlinear state space model is established with respect to shaft current, turbine rotational speed and power output in the wind energy conversion [...] Read more.
In this paper, a wind energy conversion system is studied to improve the conversion efficiency and maximize power output. Firstly, a nonlinear state space model is established with respect to shaft current, turbine rotational speed and power output in the wind energy conversion system. As the wind velocity can be descried as a non-Gaussian variable on the system model, the survival information potential is adopted to measure the uncertainty of the stochastic tracking error between the actual wind turbine rotation speed and the reference one. Secondly, to minimize the stochastic tracking error, the control input is obtained by recursively optimizing the performance index function which is constructed with consideration of both survival information potential and control input constraints. To avoid those complex probability formulation, a data driven method is adopted in the process of calculating the survival information potential. Finally, a simulation example is given to illustrate the efficiency of the proposed maximum power point tracking control method. The results demonstrate that by following this method, the actual wind turbine rotation speed can track the reference speed with less time, less overshoot and higher precision, and thus the power output can still be guaranteed under the influence of non-Gaussian wind noises. Full article
(This article belongs to the Topic Advances in Nonlinear Dynamics: Methods and Applications)
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17 pages, 410 KiB  
Article
Robust Trajectory Tracking Control for Continuous-Time Nonlinear Systems with State Constraints and Uncertain Disturbances
by Chunbin Qin, Xiaopeng Qiao, Jinguang Wang and Dehua Zhang
Entropy 2022, 24(6), 816; https://doi.org/10.3390/e24060816 - 11 Jun 2022
Cited by 3 | Viewed by 2069
Abstract
In this paper, a robust trajectory tracking control method with state constraints and uncertain disturbances on the ground of adaptive dynamic programming (ADP) is proposed for nonlinear systems. Firstly, the augmented system consists of the tracking error and the reference trajectory, and the [...] Read more.
In this paper, a robust trajectory tracking control method with state constraints and uncertain disturbances on the ground of adaptive dynamic programming (ADP) is proposed for nonlinear systems. Firstly, the augmented system consists of the tracking error and the reference trajectory, and the tracking control problems with uncertain disturbances is described as the problem of robust control adjustment. In addition, considering the nominal system of the augmented system, the guaranteed cost tracking control problem is transformed into the optimal control problem by using the discount coefficient in the nominal system. A new safe Hamilton–Jacobi–Bellman (HJB) equation is proposed by combining the cost function with the control barrier function (CBF), so that the behavior of violating the safety regulations for the system states will be punished. In order to solve the new safe HJB equation, a critic neural network (NN) is used to approximate the solution of the safe HJB equation. According to the Lyapunov stability theory, in the case of state constraints and uncertain disturbances, the system states and the parameters of the critic neural network are guaranteed to be uniformly ultimately bounded (UUB). At the end of this paper, the feasibility of the proposed method is verified by a simulation example. Full article
(This article belongs to the Topic Advances in Nonlinear Dynamics: Methods and Applications)
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23 pages, 602 KiB  
Article
Robust Tracking Control for Non-Zero-Sum Games of Continuous-Time Uncertain Nonlinear Systems
by Chunbin Qin, Ziyang Shang, Zhongwei Zhang, Dehua Zhang and Jishi Zhang
Mathematics 2022, 10(11), 1904; https://doi.org/10.3390/math10111904 - 02 Jun 2022
Cited by 5 | Viewed by 1876
Abstract
In this paper, a new adaptive critic design is proposed to approximate the online Nash equilibrium solution for the robust trajectory tracking control of non-zero-sum (NZS) games for continuous-time uncertain nonlinear systems. First, the augmented system was constructed by combining the tracking error [...] Read more.
In this paper, a new adaptive critic design is proposed to approximate the online Nash equilibrium solution for the robust trajectory tracking control of non-zero-sum (NZS) games for continuous-time uncertain nonlinear systems. First, the augmented system was constructed by combining the tracking error and the reference trajectory. By modifying the cost function, the robust tracking control problem was transformed into an optimal tracking control problem. Based on adaptive dynamic programming (ADP), a single critic neural network (NN) was applied for each player to solve the coupled Hamilton–Jacobi–Bellman (HJB) equations approximately, and the obtained control laws were regarded as the feedback Nash equilibrium. Two additional terms were introduced in the weight update law of each critic NN, which strengthened the weight update process and eliminated the strict requirements for the initial stability control policy. More importantly, in theory, through the Lyapunov theory, the stability of the closed-loop system was guaranteed, and the robust tracking performance was analyzed. Finally, the effectiveness of the proposed scheme was verified by two examples. Full article
(This article belongs to the Topic Advances in Nonlinear Dynamics: Methods and Applications)
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