Mathematical Methods in the Applied Sciences最新文献

{"title":"Normal Form Formulae of Turing–Turing Bifurcation for Partial Functional Differential Equations With Nonlinear Diffusion","authors":"Yue Xing,&nbsp;Weihua Jiang","doi":"10.1002/mma.10627","DOIUrl":"https://doi.org/10.1002/mma.10627","url":null,"abstract":"<div>\u0000 \u0000 <p>For most systems, the appearance of Turing bifurcation means that it is possible to excite Turing–Turing bifurcation, thus inducing superimposed spatial patterns, multistable spatial patterns co-existing, and others.It is undoubtedly of interest to qualitatively analyze the structures and stability of these spatially heterogeneous solutions generated by Turing–Turing bifurcation. Therefore, in this paper, with the aid of the center manifold theory and the normal form method, for general partial functional differential equations with nonlinear diffusion, the third-order normal form is firstly derived. It is locally topologically equivalent to the primitive partial functional differential equations at the Turing–Turing bifurcation point. And then the explicit formulae for the coefficients in the normal form associated with three different spatial modes are presented. As an application, a predator–prey model with predator–taxis is considered. It is theoretically revealed that the system admits the coexistence of a pair of stable steady states with a single characteristic wavelength and the coexistence of four stable steady states with different single characteristic wavelengths. Further, the parameter regions in which these phenomena will arise are quantitatively given. It is illustrated that the self-diffusion of prey and predator–taxis complement each other to promote the formation for the spatially homogeneous distributions of populations.</p>\u0000 </div>","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":"48 5","pages":"5660-5680"},"PeriodicalIF":2.1,"publicationDate":"2025-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143595522","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Unified Predefined-Time Stability Theorem and Sliding Mode Control for Fractional-Order Nonlinear Systems","authors":"Jingang Liu,&nbsp;Ruiqi Li","doi":"10.1002/mma.10634","DOIUrl":"https://doi.org/10.1002/mma.10634","url":null,"abstract":"<div>\u0000 \u0000 <p>This paper proposes a class of predefined-time stability (PDTS) theorem and a fractional-order (FO) sliding mode control algorithm for the synchronization of FO nonlinear systems. Based on the simple tetration function, a new PDTS theorem and an FO sliding mode controller are established. Then, a new unified PDTS theorem is defined, which extends the existing Lyapunov function and provides a detailed mathematical proof. The Lyapunov function is used to construct a unified FO sliding mode control algorithm, and the mathematical proof that it satisfies the PDTS is given. Finally, the proposed framework is applied to the synchronization of two different chaotic systems, and numerical simulations prove that the unified approach can extend the existing PDTS theorems and is universal and effective.</p>\u0000 </div>","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":"48 5","pages":"5755-5767"},"PeriodicalIF":2.1,"publicationDate":"2025-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143595590","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The Solvability and Sensitivity of Nonautonomous Fractional Differential Inclusions Steered by Mixed Brownian Motion","authors":"Surendra Kumar,&nbsp;Anjali Upadhyay","doi":"10.1002/mma.10712","DOIUrl":"https://doi.org/10.1002/mma.10712","url":null,"abstract":"<div>\u0000 \u0000 <p>The qualitative study of stochastic fractional nonautonomous systems in infinite-dimensional spaces is rarely available in the literature. Our aim in this article is to examine the existence and sensitivity of a mild solution for a novel class of nonautonomous stochastic fractional differential inclusions driven by standard and fractional Brownian motion (fBm). We derive the existence and uniqueness of the solution for the considered system with the help of operators generated by the probability density function and the family of linear, closed operators by utilizing the Picard iteration method. Furthermore, we investigate the sensitivity of the mild solution concerning the initial condition. The obtained results are proved under weaker assumptions than the Lipschitz conditions on the system parameters. We also use the Jensen, the Gronwall, and the Bihari inequalities to acquire the results. Our results generalize the work corresponding to fractional autonomous and deterministic systems. Lastly, two examples are constructed to validate the produced results.</p>\u0000 </div>","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":"48 6","pages":"6749-6763"},"PeriodicalIF":2.1,"publicationDate":"2025-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143622487","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Extinction, Ultimate Boundedness, and Persistence in the Mean of a Stochastic Heroin Epidemic Model With Distributed Delay","authors":"Xiaofeng Zhang","doi":"10.1002/mma.10698","DOIUrl":"https://doi.org/10.1002/mma.10698","url":null,"abstract":"<div>\u0000 \u0000 <p>In this paper, we consider a stochastic heroin epidemic model with distributed delay. We analyze the model in detail: We prove the existence and uniqueness of the global positive solution of the system, the asymptotic behavior around the equilibrium point \u0000<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mrow>\u0000 <mi>E</mi>\u0000 </mrow>\u0000 <mrow>\u0000 <mn>0</mn>\u0000 </mrow>\u0000 </msub>\u0000 </mrow>\u0000 <annotation>$$ {E}_0 $$</annotation>\u0000 </semantics></math> of the deterministic system, the stochastically ultimate boundedness, and the persistence in the mean of the disease. Finally, we verify the main conclusions of this paper through numerical simulation and explore the influence of system parameters on the persistence and extinction of diseases. According to the numerical simulation results, we can give some suggestions on controlling diseases.</p>\u0000 </div>","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":"48 6","pages":"6592-6606"},"PeriodicalIF":2.1,"publicationDate":"2025-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143622486","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A Family of Effective Spectral CG Methods With an Adaptive Restart Scheme","authors":"Haiyan Zheng,&nbsp;Xiaping Zeng,&nbsp;Pengjie Liu","doi":"10.1002/mma.10653","DOIUrl":"https://doi.org/10.1002/mma.10653","url":null,"abstract":"<div>\u0000 \u0000 <p>In this paper, we propose a family of effective spectral conjugate gradient methods with an adaptive restart scheme for solving unconstrained optimization problems. First, we construct a new composite conjugate parameter with two parameters by employing a convex combination of classical conjugate parameters and their variants. Then, we use the spectral technique to guarantee that the search direction possesses the sufficient descent property independent of any line search. Additionally, we incorporate a new spectral gradient-based adaptive restart scheme to ensure the global convergence of the family under weak Wolfe line search and obtain iteration complexity results under Armijo line search. Finally, we conduct numerical experiments on unconstrained optimization problems and image restoration applications to demonstrate the effectiveness and practicality of the proposed family.</p>\u0000 </div>","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":"48 5","pages":"6035-6047"},"PeriodicalIF":2.1,"publicationDate":"2025-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143595591","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A Novel Robust and Predefined-Time Zeroing Neural Network Solver for Time-Varying Linear Matrix Equation","authors":"Chunhao Han,&nbsp;Jiao Xu,&nbsp;Bing Zheng","doi":"10.1002/mma.10654","DOIUrl":"https://doi.org/10.1002/mma.10654","url":null,"abstract":"<div>\u0000 \u0000 <p>This paper develops a novel robust and predefined-time zeroing neural network (RPZNN) to solve the time-varying linear matrix equation (TVLME) in real time by developing an innovative activation function with a time parameter \u0000<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mrow>\u0000 <mi>t</mi>\u0000 </mrow>\u0000 <mrow>\u0000 <mi>f</mi>\u0000 </mrow>\u0000 </msub>\u0000 </mrow>\u0000 <annotation>$$ {t}_f $$</annotation>\u0000 </semantics></math>. Different from the existing ZNN solvers with complex convergence time bounds, the RPZNN solver obtains the real-time solution of the TVLME within an arbitrarily predefined time \u0000<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mrow>\u0000 <mi>t</mi>\u0000 </mrow>\u0000 <mrow>\u0000 <mi>f</mi>\u0000 </mrow>\u0000 </msub>\u0000 </mrow>\u0000 <annotation>$$ {t}_f $$</annotation>\u0000 </semantics></math>. Moreover, the RPZNN solver can freely adjust \u0000<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mrow>\u0000 <mi>t</mi>\u0000 </mrow>\u0000 <mrow>\u0000 <mi>f</mi>\u0000 </mrow>\u0000 </msub>\u0000 </mrow>\u0000 <annotation>$$ {t}_f $$</annotation>\u0000 </semantics></math> to accommodate the requirements for various convergence rates, demonstrating its considerable flexibility. We conduct a theoretical analysis for the predefined-time convergence of the RPZNN solver and its robustness against additive noise interference. Furthermore, numerical experiments validate the effectiveness of the RPZNN in accurately addressing the TVLME and demonstrate its superior performance in terms of convergence rate and robustness when compared to several traditional or state-of-the-art ZNN solvers. Additionally, the RPZNN solver also exhibits excellent capabilities in dynamic alternating current (DAC) computing and the 6-link planar robot manipulator (6PRM) path-tracking task, highlighting its potential for wide-ranging applications.</p>\u0000 </div>","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":"48 5","pages":"6048-6062"},"PeriodicalIF":2.1,"publicationDate":"2025-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143595518","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Excess Risk Bound for Deep Learning Under Weak Dependence","authors":"William Kengne","doi":"10.1002/mma.10719","DOIUrl":"https://doi.org/10.1002/mma.10719","url":null,"abstract":"<div>\u0000 \u0000 <p>This paper considers deep neural networks for learning weakly dependent processes in a general framework that includes, for instance, regression estimation, time series prediction, time series classification. The \u0000<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>ψ</mi>\u0000 </mrow>\u0000 <annotation>$$ psi $$</annotation>\u0000 </semantics></math>-weak dependence structure considered is quite large and covers other conditions such as mixing, association, and so on. Firstly, the approximation of smooth functions by deep neural networks with a broad class of activation functions is considered. We derive the required depth, width and sparsity of a deep neural network to approximate any Hölder smooth function, defined on any compact set \u0000<span></span><math>\u0000 <mrow>\u0000 <mi>𝒳</mi>\u0000 </mrow></math>. Secondly, we establish a bound of the excess risk for the learning of weakly dependent observations by deep neural networks. When the target function is sufficiently smooth, this bound is close to the usual \u0000<span></span><math>\u0000 <mrow>\u0000 <mi>𝒪</mi>\u0000 <mo>(</mo>\u0000 <msup>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 <mrow>\u0000 <mo>−</mo>\u0000 <mn>1</mn>\u0000 <mo>/</mo>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 </msup>\u0000 <mo>)</mo>\u0000 </mrow></math>.</p>\u0000 </div>","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":"48 6","pages":"6844-6850"},"PeriodicalIF":2.1,"publicationDate":"2025-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143622485","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Dynamics of a Two-Strain Model With Vaccination, General Incidence Rate, and Nonlocal Diffusion","authors":"Arturo J. Nic-May,&nbsp;Eric J. Avila-Vales","doi":"10.1002/mma.10680","DOIUrl":"https://doi.org/10.1002/mma.10680","url":null,"abstract":"<div>\u0000 \u0000 <p>In this paper, we studied a two-strain model with vaccination, general incidence rate, and nonlocal diffusion. In this model, it is considered that those vaccinated can become infected with either strain and that those removed individuals with respect to strain 1 can become infected with strain 2 (partial cross-immunity). We prove that solution exists and is unique, bounded, and positive, and there exists a disease-free steady state. The basic reproduction number \u0000<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mrow>\u0000 <mi>ℛ</mi>\u0000 </mrow>\u0000 <mrow>\u0000 <mn>0</mn>\u0000 </mrow>\u0000 </msub>\u0000 </mrow>\u0000 <annotation>$$ {mathcal{R}}_0 $$</annotation>\u0000 </semantics></math> is defined, and the existence of principal eigenvalue is studied. For \u0000<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mrow>\u0000 <mi>ℛ</mi>\u0000 </mrow>\u0000 <mrow>\u0000 <mn>0</mn>\u0000 </mrow>\u0000 </msub>\u0000 <mo>&lt;</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 <annotation>$$ {mathcal{R}}_0&amp;lt;1 $$</annotation>\u0000 </semantics></math>, we prove the disease-free steady state is globally asymptotically stable, and when \u0000<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mrow>\u0000 <mi>ℛ</mi>\u0000 </mrow>\u0000 <mrow>\u0000 <mn>0</mn>\u0000 </mrow>\u0000 </msub>\u0000 <mo>&gt;</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 <annotation>$$ {mathcal{R}}_0&amp;gt;1 $$</annotation>\u0000 </semantics></math>, the disease-free steady state is unstable. We also prove the uniform persistence of the system. Four steady states were obtained, and it was shown that the existence of each of the steady state depends on 4 threshold quantities and we used a Lyapunov approach to prove the global stability of the steady state under certain assumptions. The results are confirmed using some graphical representations.</p>\u0000 </div>","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":"48 6","pages":"6396-6424"},"PeriodicalIF":2.1,"publicationDate":"2025-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143622267","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Dynamic Behaviors and Bifurcation Analysis of a Three-Dimensional Filippov Ecosystem With Fear Effect","authors":"Mengting Hu,&nbsp;Changcheng Xiang,&nbsp;Ben Chu","doi":"10.1002/mma.10699","DOIUrl":"https://doi.org/10.1002/mma.10699","url":null,"abstract":"<div>\u0000 \u0000 <p>A Filippov system of crop-pest-natural enemy with a Holling-II type functional response function is developed based on the fear effect and threshold control strategy. The threshold control strategy is mainly a means of controlling pests and natural enemies. When the number of natural enemies is below a threshold, the natural enemies cannot achieve a controlling effect on the pest population and control is needed to suppress the outbreak. In this case, the fear of pests to natural enemies will affect the pest's survival mode, thus affecting the pest's dynamic behavior. However, the number of natural enemy population exceeds the threshold density, and there is no need to control the system. At this point, the pest's fear of natural enemies is almost zero. The dynamic behavior of the two subsystems of the model is discussed, the existence and stability of various equilibria are analyzed, and the existence of sliding and crossing regions is also investigated. In addition, it is possible that the established model has more than one pseudo-equilibrium. Therefore, the dynamic behavior of the pseudo-equilibria is analyzed, and we observe that Hopf bifurcation occurs near the pseudo-equilibria. By numerically simulating the global sliding bifurcation of the system, we discover that as the bifurcation parameters are varied, the system exhibits a series of bifurcations such as grazing bifurcation, buckling bifurcation, crossing bifurcation, and period-halving bifurcation.</p>\u0000 </div>","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":"48 6","pages":"6607-6623"},"PeriodicalIF":2.1,"publicationDate":"2025-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143622309","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Relative Controllability of Fractional Dynamical Systems With a Delay in State and Multiple Delays in Control","authors":"Mustafa Aydin,&nbsp;Nazim I. Mahmudov","doi":"10.1002/mma.10714","DOIUrl":"https://doi.org/10.1002/mma.10714","url":null,"abstract":"<div>\u0000 \u0000 <p>This work is devoted to the study of the relative controllability of fractional dynamic systems in finite-dimensional spaces with a state delay and multiple delays in control. For linear systems to be relatively controllable, necessary and sufficient conditions are determined by defining and using the Gramian matrix. The controllability conditions for semilinear systems are determined on the basis of Schauder's fixed point theorem.</p>\u0000 </div>","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":"48 6","pages":"6779-6787"},"PeriodicalIF":2.1,"publicationDate":"2025-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143622707","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}