数学最新文献

{"title":"The central limit theorems for integrable Hamiltonian systems perturbed by white noise","authors":"","doi":"10.1016/j.jde.2024.09.047","DOIUrl":"10.1016/j.jde.2024.09.047","url":null,"abstract":"<div><div>In this paper, we consider the dynamics of integrable stochastic Hamiltonian systems. Utilizing the Nagaev-Guivarc'h method, we obtain several generalized results of the central limit theorem. Making use of this technique and the Birkhoff ergodic theorem, we prove that the invariant tori persist under stochastic perturbations. Moreover, they asymptotically follow a Gaussian distribution, which gives a positive answer to the stability of integrable stochastic Hamiltonian systems over time. Our results hold true for both Gaussian and non-Gaussian noises, and their intensities can be not small.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":2.4,"publicationDate":"2024-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142326437","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A hybrid lattice Boltzmann and finite difference method for two-phase flows with soluble surfactants","authors":"","doi":"10.1016/j.camwa.2024.09.022","DOIUrl":"10.1016/j.camwa.2024.09.022","url":null,"abstract":"<div><div>A hybrid method is developed to simulate two-phase flows with soluble surfactants. In this method, the interface and bulk surfactant concentration equations of diffuse-interface form, which include source terms to consider surfactant adsorption and desorption dynamics, are solved in the entire fluid domain by the finite difference method, while two-phase flows are solved by a lattice Boltzmann color-gradient model, which can accurately simulate binary fluids with unequal densities. The flow and interface surfactant concentration fields are coupled by a modified Langmuir equation of state, which allows for surfactant concentration beyond critical micelle concentration. The capability and accuracy of the hybrid method are first validated by simulating three numerical examples, including the adsorption of bulk surfactants onto the interface of a stationary droplet, the droplet migration in a constant surfactant gradient, and the deformation of a surfactant-laden droplet in a simple shear flow, in which the numerical results are compared with theoretical solutions and available literature data. Then, the hybrid method is applied to simulate the buoyancy-driven bubble rise in a surfactant solution, in which the influence of surfactants is identified for varying wall confinement, density ratio, Eotvos number and Biot number. It is found that surfactants exhibit a retardation effect on the bubble rise due to the Marangoni stress that resists interface motion, and the retardation effect weakens as the Eotvos or Biot number increases. We further show that the weakened retardation effect at higher Biot numbers is attributed to a decreased non-uniform effect of surfactants at the interface. By comparing with the Cahn-Hilliard phase-field method, we also show that the present method conserves the mass for each fluid, improves numerical stability especially at high density ratio and Eotvos number, and does not need the selection of free parameters, thus breaking the limitations of the existing method.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142327241","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Analysis of rogue wave in the mid-infrared supercontinuum under femtosecond weak seed pulse conditions based on deep learning","authors":"","doi":"10.1016/j.chaos.2024.115575","DOIUrl":"10.1016/j.chaos.2024.115575","url":null,"abstract":"<div><div>The generation process of rogue wave (RW) is affected by noise, which is an unstable state, and the existence of RW will reduce the stability of mid-infrared supercontinuum. However, the process of studying RW requires a large amount of data simulation and statistics, and traditional methods are time-consuming and inefficient. Therefore, this paper adopts long short-term memory (LSTM) neural network to obtain the spectrum information after transmission for a certain distance according to the waveform information of the incident pulse. The results show that the LSTM neural network structure can train and predict the peak power, time deviation information, time intensity evolution and spectrum evolution of RW after 10 cm propagation with only changing the number of internal units. And it performs well on both large and small data sets.</div></div>","PeriodicalId":9764,"journal":{"name":"Chaos Solitons & Fractals","volume":null,"pages":null},"PeriodicalIF":5.3,"publicationDate":"2024-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142327874","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Arithmetic Demailly approximation theorem","authors":"","doi":"10.1016/j.aim.2024.109961","DOIUrl":"10.1016/j.aim.2024.109961","url":null,"abstract":"<div><div>We generalize the Demailly approximation theorem from complex geometry to Arakelov geometry.</div><div>As an application, let <span><math><mi>X</mi><mo>/</mo><mi>Q</mi></math></span> be an integral projective variety and <span><math><mover><mrow><mi>N</mi></mrow><mo>‾</mo></mover></math></span> be an adelic line bundle on <em>X</em>. We prove that <span><math><mi>ess</mi><mo>(</mo><mover><mrow><mi>N</mi></mrow><mo>‾</mo></mover><mo>)</mo><mo>≥</mo><mn>0</mn></math></span> ⟹ <span><math><mover><mrow><mi>N</mi></mrow><mo>‾</mo></mover></math></span> pseudo-effective. This was proved in <span><span>[1]</span></span>, assuming <span><math><mover><mrow><mi>N</mi></mrow><mo>‾</mo></mover></math></span> relatively semipositive.</div><div>We show in the appendix that the above assertion is also true for adelic line bundles on quasi-projective varieties, under the framework of <span><span>[17]</span></span>.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142328281","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Covariance Assisted Multivariate Penalized Additive Regression (CoMPAdRe)","authors":"Neel Desai, Veerabhadran Baladandayuthapani, Russell T. Shinohara, Jeffrey S. Morris","doi":"10.1080/10618600.2024.2407453","DOIUrl":"https://doi.org/10.1080/10618600.2024.2407453","url":null,"abstract":"We propose a new method for the simultaneous selection and estimation of multivariate sparse additive models with correlated errors. Our method called Covariance Assisted Multivariate Penalized Add...","PeriodicalId":15422,"journal":{"name":"Journal of Computational and Graphical Statistics","volume":null,"pages":null},"PeriodicalIF":2.4,"publicationDate":"2024-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142328965","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Optimized five-by-five block preconditioning for efficient GMRES convergence in curvature-based image deblurring","authors":"","doi":"10.1016/j.camwa.2024.09.026","DOIUrl":"10.1016/j.camwa.2024.09.026","url":null,"abstract":"<div><div>We introduce an enhanced preconditioning technique designed to expedite the convergence of Krylov subspace methods when dealing with non-linear systems of equations featuring a block five-by-five format. This scenario often arises in the context of cell centered finite difference discretizations applied to the mean curvature based image deblurring problem. A thorough spectral analysis of the preconditioned matrices reveals a favorable eigenvalue distribution, leading to accelerated convergence of preconditioned Generalized Minimal Residual (GMRES) methods. Furthermore, we present numerical experiments to showcase the effectiveness of this preconditioner when combined with the flexible GMRES solver for addressing non-linear systems of equations originating from a image deblurring problems. Codes can be obtained at <span><span>https://github.com/shahbaz1982/Preconditioning</span><svg><path></path></svg></span>.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142326369","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A topological approach to mapping space signatures","authors":"","doi":"10.1016/j.aam.2024.102787","DOIUrl":"10.1016/j.aam.2024.102787","url":null,"abstract":"<div><div>A common approach for describing classes of functions and probability measures on a topological space <span><math><mi>X</mi></math></span> is to construct a suitable map Φ from <span><math><mi>X</mi></math></span> into a vector space, where linear methods can be applied to address both problems. The case where <span><math><mi>X</mi></math></span> is a space of paths <span><math><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo><mo>→</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> and Φ is the path signature map has received much attention in stochastic analysis and related fields. In this article we develop a generalized Φ for the case where <span><math><mi>X</mi></math></span> is a space of maps <span><math><msup><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow><mrow><mi>d</mi></mrow></msup><mo>→</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> for any <span><math><mi>d</mi><mo>∈</mo><mi>N</mi></math></span>, and show that the map Φ generalizes many of the desirable algebraic and analytic properties of the path signature to <span><math><mi>d</mi><mo>≥</mo><mn>2</mn></math></span>. The key ingredient to our approach is topological; in particular, our starting point is a generalization of K-T Chen's path space cochain construction to the setting of cubical mapping spaces.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142326570","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 Latent Space Model for Weighted Keyword Co-occurrence Networks with Applications in Knowledge Discovery in Statistics","authors":"Yan Zhang, Rui Pan, Xuening Zhu, Kuangnan Fang, Hansheng Wang","doi":"10.1080/10618600.2024.2407465","DOIUrl":"https://doi.org/10.1080/10618600.2024.2407465","url":null,"abstract":"Keywords are widely recognized as pivotal in conveying the central idea of academic articles. In this article, we construct a weighted and dynamic keyword co-occurrence network and propose a latent...","PeriodicalId":15422,"journal":{"name":"Journal of Computational and Graphical Statistics","volume":null,"pages":null},"PeriodicalIF":2.4,"publicationDate":"2024-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142328964","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Mass conservation in the validation of fluid-poroelastic structure interaction solvers","authors":"","doi":"10.1016/j.amc.2024.129081","DOIUrl":"10.1016/j.amc.2024.129081","url":null,"abstract":"<div><div>Benchmark problems commonly used to test numerical methods for fluid-poroelastic structure interaction often rely on simple examples constructed using the method of manufactured solutions. In this work, we show that such examples are not adequate to demonstrate the performance of the method, especially in cases when the poroelastic system is written in the primal or primal-mixed formulation, and when the dynamics of the poroelastic structure are driven only by dynamic loading from the fluid, which often occurs in biomedical applications. In those cases, the only forcing on the structure comes from the interaction with the fluid at the fluid-structure interface, where the coupling conditions are imposed. One of those conditions is a kinematic condition which enforces the conservation of mass. If this condition is not accurately satisfied, the resulting dynamics might lead to highly inaccurate results in the entire domain. We present three benchmark problems: Example 1 is based on the method of manufactured solutions; Example 2 is based on parameters used in geomechanics; and Example 3 is a benchmark problem with parameters from hemodynamics. Using these examples, we test the performance of the primal, primal-mixed and dual-mixed formulations. While all methods perform well in the first two examples, the primal and primal-mixed formulations exhibit large errors in Example 3, where the densities of the fluid and solid are comparable, and the structure dynamics is purely driven by the fluid loading. To recover the accuracy, we propose to use the primal and primal-mixed methods with a penalty term, which helps to enforce the conservation of mass.</div></div>","PeriodicalId":55496,"journal":{"name":"Applied Mathematics and Computation","volume":null,"pages":null},"PeriodicalIF":3.5,"publicationDate":"2024-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142328083","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Bifurcation and dynamics of periodic solutions of MEMS model with squeeze film damping","authors":"","doi":"10.1016/j.nonrwa.2024.104229","DOIUrl":"10.1016/j.nonrwa.2024.104229","url":null,"abstract":"&lt;div&gt;&lt;div&gt;In this paper, we study the oscillations of an idealized mass–spring model of micro-electro-mechanical system (MEMS) with squeeze film damping. The model consists of two parallel electrodes separated by a gap &lt;span&gt;&lt;math&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;: one of them is fixed, and another one is movable and attached to a linear spring with stiffness coefficient &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;. The oscillation, under the influence of AC–DC voltage &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;cos&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;π&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, is ruled by the following singular differential equation &lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mrow&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/mrow&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;θ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;Here, &lt;span&gt;&lt;math&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; is the vertical displacement of the moving plate (&lt;span&gt;&lt;math&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; is always assumed to be less than &lt;span&gt;&lt;math&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;), &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; is its mass, &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; is the electrode area, and &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;θ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; is the absolute dielectric constant of vacuum. Taking &lt;span&gt;&lt;math&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; as the parameter, we show the existence of saddle–node bifurcation of &lt;span&gt;&lt;math&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;-periodic solutions to the equation in the parameter space. This answers, from certain point of view, the open problem proposed by Torres in his monograph, see Torres (2015, Open Problem 2.1, p. 18). Further, we prove that the equation has exactly two classes of &lt;span&gt;&lt;math&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;-periodic solutions: as &lt;span&gt;&lt;math&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; tends to &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;∞&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, one of them uniformly tends t","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2024-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142328262","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}