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Mathematics
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Research on Dynamical Systems and Differential Equations
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Research on Dynamical Systems and Differential Equations

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "C1: Difference and Differential Equations".

Deadline for manuscript submissions: closed (30 April 2025) | Viewed by 5381

Special Issue Editors

Dr. Ahmadjan Muhammadhaji

E-Mail Website
Guest Editor
1. College of Mathematics and System Sciences, Xinjiang University, Urumqi 830017, China
2. The Key Laboratory of Applied Mathematics of Xinjiang Uygur Autonomous Region, Xinjiang University, Urumqi 830017, China
Interests: differential equation; dynamical systems
Dr. Maimaiti Yimamu

E-Mail Website
Guest Editor
College of Mathematics and System Sciences, Xinjiang University, Urumqi 830017, China
Interests: differential equation; dynamical systems

Special Issue Information

Dear Colleagues,

Dynamical systems and differential equations are fundamental areas of mathematics that have been applied in a wide range of fields, including physics, chemistry, engineering, biology, economics, electronics, ecology, epidemiology, neural networks, and many other real-world applications. Dynamical systems refer to a collection of mathematical models that describe the time evolution of physical or abstract systems. Differential equations, on the other hand, are mathematical equations that describe the relationship between a function and its derivatives.

Overall, dynamical systems and differential equations comprise a vibrant and active field with numerous applications and exciting open problems.

Therefore, this Special Issue aims to collate findings of mathematicians, biologists, physicists, epidemiologists, engineers, economists, ecologists, mechanists, and other scientists for whom dynamical systems and differential equations are valuable research tools.

Dr. Ahmadjan Muhammadhaji
Dr. Maimaiti Yimamu
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

  • differential equations
  • deterministic and stochastic differential equations
  • dynamical systems
  • modeling and dynamics in population dynamical systems
  • complex networks
  • neural networks
  • epidemic models

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Published Papers (6 papers)

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Research

26 pages, 2256 KiB  
Article
A Rumor-Spreading Model with Three Identical Time Delays
by Chunlong Fu, Guofang Liu, Xiaofan Yang, Yang Qin and Luxing Yang
Mathematics 2025, 13(9), 1421; https://doi.org/10.3390/math13091421 - 26 Apr 2025
Viewed by 91
Abstract
Understanding the effect of time delays on rumor spreading is of special importance to curbing the spread of rumors. This article proposes a rumor-spreading model with three identical time delays: a delay associated with the negative influence of a spreader on an exposed [...] Read more.
Understanding the effect of time delays on rumor spreading is of special importance to curbing the spread of rumors. This article proposes a rumor-spreading model with three identical time delays: a delay associated with the negative influence of a spreader on an exposed ignorant individual, a delay associated with the natural change from a spreader to a stifler, and a delay associated with the positive influence of a stifler on an exposed spreader. The basic reproduction number for the model is determined. A criterion for the existence of rumor-endemic equilibrium is provided. Interestingly, the model undergoes a conditional forward bifurcation. A collection of criteria for the asymptotic stability of the rumor-free equilibrium is derived. In the absence of a time delay, a criterion for the asymptotic stability of the rumor-endemic equilibrium is presented. By developing a novel technique for dealing with small time delays, a criterion for the asymptotic stability of the rumor-endemic equilibrium is established. Finally, the effect of some factors on the existence of rumor-endemic equilibrium is investigated. In particular, the effect of the time delay on rumor spreading is revealed. This work facilitates a deep understanding of the dynamics of rumor-spreading models with time delays. Full article
(This article belongs to the Special Issue Research on Dynamical Systems and Differential Equations)
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34 pages, 885 KiB  
Article
Dynamic Analysis of a Standby System with Retrial Strategies and Multiple Working Vacations
by Changjiang Lai, Ehmet Kasim and Ahmadjan Muhammadhaji
Mathematics 2024, 12(24), 3999; https://doi.org/10.3390/math12243999 - 19 Dec 2024
Cited by 1 | Viewed by 766
Abstract
In this paper, we developed a new standby system that combines a retrial strategy with multiple working vacations, and we performed a dynamic analysis of the system. We investigated its well−posedness and asymptotic behavior using the theory of the C0−semigroup [...] Read more.
In this paper, we developed a new standby system that combines a retrial strategy with multiple working vacations, and we performed a dynamic analysis of the system. We investigated its well−posedness and asymptotic behavior using the theory of the C0−semigroup in the functional analysis. First, the corresponding model was transformed into an abstract Cauchy problem in Banach space by introducing the state space, the main operator, and its domain of definition. Second, we demonstrated that the model had a unique non−negative time−dependent solution. Using Greiner’s boundary perturbation idea and the spectral properties of the corresponding operator, the non−negative time−dependent solution strongly converged to its steady−state solution. We also provide numerical examples to illustrate the effect of different parameters on the system’s reliability metrics. Full article
(This article belongs to the Special Issue Research on Dynamical Systems and Differential Equations)
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20 pages, 6617 KiB  
Article
Fault Monitoring Method for the Process Industry System Based on the Improved Dense Connection Network
by Jiarula Yasenjiang, Zhigang Lan, Kai Wang, Luhui Lv, Chao He, Yingjun Zhao, Wenhao Wang and Tian Gao
Mathematics 2024, 12(18), 2843; https://doi.org/10.3390/math12182843 - 13 Sep 2024
Viewed by 685
Abstract
The safety of chemical processes is of critical importance. However, traditional fault monitoring methods have insufficiently studied the monitoring accuracy of multi-channel data and have not adequately considered the impact of noise on industrial processes. To address this issue, this paper proposes a [...] Read more.
The safety of chemical processes is of critical importance. However, traditional fault monitoring methods have insufficiently studied the monitoring accuracy of multi-channel data and have not adequately considered the impact of noise on industrial processes. To address this issue, this paper proposes a neural network-based model, DSCBAM-DenseNet, which integrates depthwise separable convolution and attention modules to fuse multi-channel data features and enhance the model’s noise resistance. We simulated a real environment by adding Gaussian noise with different signal-to-noise ratios to the Tennessee Eastman process dataset and trained the model using multi-channel data. The experimental results show that this model outperforms traditional models in both fault diagnosis accuracy and noise resistance. Further research on a compressor unit engineering instance validated the superiority of the model. Full article
(This article belongs to the Special Issue Research on Dynamical Systems and Differential Equations)
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20 pages, 339 KiB  
Article
Local C0,1-Regularity for the Parabolic p-Laplacian Equation on the Group SU(3)
by Yongming He, Chengwei Yu and Hongqing Wang
Mathematics 2024, 12(9), 1288; https://doi.org/10.3390/math12091288 - 24 Apr 2024
Viewed by 1183
Abstract
In this article, when 2≤p≤4, we establish the Cloc0,1-regularity of weak solutions to the degenerate parabolic p-Laplacian equation [...] Read more.
In this article, when 2≤p≤4, we establish the Cloc0,1-regularity of weak solutions to the degenerate parabolic p-Laplacian equation ∂tu=−∑i=16Xi*(|∇Hu|p−2Xiu) on the group SU(3) granted with horizontal vector fields X1, …, X6. Compared to the Heisenberg group, Hn, we obtained the optimal range of p; that is, 2≤p≤4. Full article
(This article belongs to the Special Issue Research on Dynamical Systems and Differential Equations)
19 pages, 8020 KiB  
Article
Stochastic Synchronization of Impulsive Reaction–Diffusion BAM Neural Networks at a Fixed and Predetermined Time
by Rouzimaimaiti Mahemuti, Ehmet Kasim and Hayrengul Sadik
Mathematics 2024, 12(8), 1204; https://doi.org/10.3390/math12081204 - 17 Apr 2024
Viewed by 996
Abstract
This paper discusses the synchronization problem of impulsive stochastic bidirectional associative memory neural networks with a diffusion term, specifically focusing on the fixed-time (FXT) and predefined-time (PDT) synchronization. First, a number of more relaxed lemmas are introduced for the FXT and PDT stability [...] Read more.
This paper discusses the synchronization problem of impulsive stochastic bidirectional associative memory neural networks with a diffusion term, specifically focusing on the fixed-time (FXT) and predefined-time (PDT) synchronization. First, a number of more relaxed lemmas are introduced for the FXT and PDT stability of general types of impulsive nonlinear systems. A controller that does not require a sign function is then proposed to ensure that the synchronization error converges to zero within a predetermined time. The controllerdesigned in this paper serves the additional purpose of preventing the use of an unreliable inequality in the course of proving the main results. Next, to guarantee FXT and PDT synchronization of the drive–response systems, this paper employs the Lyapunov function method and derives sufficient conditions. Finally, a numerical simulation is presented to validate the theoretical results. Full article
(This article belongs to the Special Issue Research on Dynamical Systems and Differential Equations)
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26 pages, 366 KiB  
Article
Dynamic Analysis of the M/G/1 Stochastic Clearing Queueing Model in a Three-Phase Environment
by Nurehemaiti Yiming
Mathematics 2024, 12(6), 805; https://doi.org/10.3390/math12060805 - 8 Mar 2024
Cited by 1 | Viewed by 824
Abstract
In this paper, we consider the M/G/1 stochastic clearing queueing model in a three-phase environment, which is described by integro-partial differential equations (IPDEs). Our first result is semigroup well-posedness for the dynamic system. Utilizing a C0—semigroup theory, we prove that the [...] Read more.
In this paper, we consider the M/G/1 stochastic clearing queueing model in a three-phase environment, which is described by integro-partial differential equations (IPDEs). Our first result is semigroup well-posedness for the dynamic system. Utilizing a C0—semigroup theory, we prove that the system has a unique positive time-dependent solution (TDS) that satisfies the probability condition. As our second result, we prove that the TDS of the system strongly converges to its steady-state solution (SSS) if the service rates of the servers are constants. For this asymptotic behavior, we analyze the spectrum of the system operator associated with the system. Additionally, the stability of the semigroup generated by the system operator is also discussed. Full article
(This article belongs to the Special Issue Research on Dynamical Systems and Differential Equations)
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