Global Attractor Research Articles

Global attraction to solitons for 2D Maxwell–Lorentz equations with spinning particle

The subject of the paper is the 2D Maxwell–Lorentz system that describes a rotating particle in electromagnetic field. The system admits stationary soliton-type solutions. Attraction to solitons is proved for any finite energy solution on the basis of C 2 C^2 -regularity of the resolvent.

Precise Localization for Anatomo-Physiological Hallmarks of the Cervical Spine by Using Neural Memory Ordinary Differential Equation.

In the evaluation of cervical spine disorders, precise positioning of anatomo-physiological hallmarks is fundamental for calculating diverse measurement metrics. Despite the fact that deep learning has achieved impressive results in the field of keypoint localization, there are still many limitations when facing medical image. First, these methods often encounter limitations when faced with the inherent variability in cervical spine datasets, arising from imaging factors. Second, predicting keypoints for only 4% of the entire X-ray image surface area poses a significant challenge. To tackle these issues, we propose a deep neural network architecture, NF-DEKR, specifically tailored for predicting keypoints in cervical spine physiological anatomy. Leveraging neural memory ordinary differential equation with its distinctive memory learning separation and convergence to a singular global attractor characteristic, our design effectively mitigates inherent data variability. Simultaneously, we introduce a Multi-Resolution Focus module to preprocess feature maps before entering the disentangled regression branch and the heatmap branch. Employing a differentiated strategy for feature maps of varying scales, this approach yields more accurate predictions of densely localized keypoints. We construct a medical dataset, SCUSpineXray, comprising X-ray images annotated by orthopedic specialists and conduct similar experiments on the publicly available UWSpineCT dataset. Experimental results demonstrate that compared to the baseline DEKR network, our proposed method enhances average precision by 2% to 3%, accompanied by a marginal increase in model parameters and the floating-point operations (FLOPs). The code (https://github.com/Zhxyi/NF-DEKR) is available.

Global attractivity of tick population dynamics model with patch structure and multiple time-varying delays

The impact of multiple time delays and the variation of populations between patches make population dynamics research more realistic and challenging. The main goal of this work is to study the effect of delay on the asymptotic behavior of tick population dynamics model with patch structure and multiple time-varying delays. We make use of the fluctuation lemma and inequality techniques to tackle the positivity, persistence and the global attractivity of the considered high-dimensional system. Finally, we give an example with three numerical simulations to verify that our theoretical results are completely new and complement some earlier existing ones. Our work is conducive to further understanding the more complex laws of tick population dynamics.

Global Attractor and Singular Limits of the 3D Voigt-regularized Magnetohydrodynamic Equations

Global Attractor and Singular Limits of the 3D Voigt-regularized Magnetohydrodynamic Equations

Global attractor for the weakly damped forced Kawahara equation on the torus

We study the long-time behaviour of solutions for the weakly damped forced Kawahara equation on the torus. More precisely, we prove the existence of a global attractor in L 2 , to which as time passes all solutions draw closer. In fact, we show that the global attractor turns out to lie in a smoother space H 2 and be bounded therein. Further, we give an upper bound of the size of the attractor in H 2 that depends only on the damping parameter and the norm of the forcing term.

International Student Mobility: Pushing-Pulling Factors for International Students

International student mobility (ISM) has been considered as a key factor for many higher education institutions (HEIs) in terms of benefits that it generates. Recent literature highlights the influence of ISM in the reconfiguration of the form of educational system making it more competitive and attractive for international students. In this regard, students are paying attention to the opportunities and benefits that an international student mobility programme, such as the Erasmus programme offers. Nevertheless, it is important to take in consideration challenges and barriers that international students have to cope during their study experience abroad. The aim of this study is to analyse which are the pushing and pulling factors for international students when choosing a country as the abroad study destination. An online survey is used to Albanian international students that has previously experienced ISM programme and are currently experiencing ISM programme abroad. Findings offer important insights on understanding factors that impact international students’ decision when choosing a foreign country as a study destination. This study offers contribution in enriching the existent literature on ISM and in particular on the decision-making of international students from a developing country perspective. The study’s implications suggest that higher education institutions can boost their global attractiveness and competitiveness by addressing the key factors influencing international students’ decisions, as identified in this study. Received: 20 July 2024 / Accepted: 20 September 2024 / Published: 05 November 2024

Review of Some Modified Generalized Korteweg–de Vries–Kuramoto–Sivashinsky Equations (Part II)

In part I of this work to appear in Foudations-MDPI 2024, some existence and uniqueness results for the solutions of some equations were reviewed, such as the Korteweg–de Vries equation (KdV), the Kuramoto–Sivashinsky equation (KS), the generalized Korteweg–de Vries–Kuramoto–Sivashinsky equation (gKdV-KS), and the nonhomogeneous boundary value problem for the KdV-KS equation in quarter plane. The main objective of this paper is to review some results of the existence of global attractors for the evolution equations with nonlinearity of the form N(ux), where ux denotes the derivative of u with respect to x, focusing in particular on the Kuramoto–Sivashinsky equation in one and two dimensions. In order to illustrate the general abstract results, we have chosen to discuss in detail the existence of global attractors for the Kuramoto–Sivashinsky (KS) equation in 1D and 2D. Once a global attractor is obtained, the question arises whether it has special regularity properties. Then we give an integrated version of the homogeneous steady state Kuramoto–Sivashinsky equation in Rn. This work ends with a change from rectangular to polar coordinates in the three-dimensional KS equation to give an energy estimate in this case.

Asymptotic behavior of the Boussinesq equation with nonlocal weak damping and arbitrary growth nonlinear function

In this paper, we consider the asymptotic behavior of the Boussinesq equation with nonlocal weak damping when the nonlinear function is arbitrary polynomial growth. We firstly prove the well‐posedness of solution by means of the monotone operator theory. At the same time, we obtain the dissipative property of the dynamical system associated with the problem in the space and , respectively. After that, the asymptotic smoothness of the dynamical system and the further quasi‐stability are demonstrated by the energy reconstruction method. Finally, we show not only the existence of the finite global attractor but also the existence of the generalized exponential attractor.

Existence of a Global Attractor for the Reaction–Diffusion Equation with Memory and Lower Regularity Terms

This paper investigates the large time behavior for the reaction–diffusion equation with memory and the forcing term g∈H−1(Ω). We prove the existence of a global attractor in L2(Ω)×Lμ2(R;H01(Ω)). Due to the lower regularity of g, one can hardly use the traditional energy estimates to derive the existence of a bounded absorbing set in the higher regularity space and then the compactness of the semigroup. Here, we utilize the contractive function method to establish the asymptotic smoothness of the semigroup.

Stability of Certain Non-Autonomous Cooperative Systems of Difference Equations with the Application to Evolutionary Dynamics

This paper investigates the dynamics of non-autonomous cooperative systems of difference equations with asymptotically constant coefficients. We are mainly interested in global attractivity results for such systems and the application of such results to evolutionary population cooperation models. We use two methods to extend the global attractivity results for autonomous cooperative systems to related non-autonomous cooperative systems which appear in recent problems in evolutionary dynamics.

Existence of global attractor in reaction–diffusion model of obesity-induced Alzheimer’s disease and its control strategies

Evidence suggests that obesity, diabetes, and aging notably increase susceptibility to dementia-related conditions such as Alzheimer’s disease (AD). This article explores the correlations between obesity, diabetes, and AD. It introduces a diffusion-driven model encompassing variables like glucose dynamics, insulin levels, beta cells, microglia, cytokines, amyloid-β plaques, neurofibrillary tangles (τ plaques), neurodegeneration, and cognitive decline. The study includes stability analysis (local and global), examining boundedness and long-time behavior via showing the existence of a global attractor for the diffusion-driven model. A global sensitivity analysis, utilizing the Partial Rank Correlation Coefficient (PRCC), identifies factors sensitively impacting Aβ plaque growth, τ plaques, and neurodegeneration. The deterministic model solution illustrates spatiotemporal dynamics, revealing a link between obesity and Alzheimer’s, which is characterized by distinct patchy patterns. While Alzheimer’s has no cure, employing optimal control techniques can help alleviate its effects and enhance affected individuals’ quality of life. An optimal control problem for AD management is developed, optimizing multiple aspects of disease management. The study highlights the efficacy of long-term healthy lifestyle practices and customized anti-amyloid therapy in significantly delaying obesity-induced AD progression. This research sheds light on the connection between obesity and Alzheimer’s, underscoring the negative impact of pro-inflammatory microglia on cognitive decline while proposing control strategies.

Species identification and pollination biology of an economically important true halophyte, Salicornia brachiata Roxb.

Members of the genus Salicornia have gained a global attraction due to their ability to thrive under high saline conditions and as potential candidates in saline agriculture. However, it has been a taxonomically challenging genus for decades since the members show plastic responses to extreme environmental conditions and due to incongruences between morphological and molecular identification methods. While only a handful of commercially grown Salicornia species are fully described, most of the species including S. brachiata, a native species in the Indian sub-continent, Myanmar, and Sri Lanka are poorly described. With the potentials in adapting S. brachiata in saline agriculture, the aim of this study was to establish a morphology and DNA barcode-based species delineation system and to study pollination biology for future crop improvement projects. Tentatively identified S. brachiata plant samples were collected from two populations in Sri Lanka and completely described. GenBank lacked authenticated barcode data for S. brachiata except for one chloroplast genome to which the matK sequence obtained in the present study matched with 100 % identity. For the first time, well defined sequences of three barcode regions, ITS, ETS and matK, of S. brachiata were made available for accurate species identification. Reproductive dynamics in different parts of the inflorescence was studied. A facultative xenogamous mating system was recorded for the first time in the genus and while the lower florets in the cladode showed a preference towards outcrossing, the upper florets displayed adaptations for selfing. Data could be effectively utilized in future Salicornia breeding programs.

Calmed 3D Navier–Stokes Equations: Global Well-Posedness, Energy Identities, Global Attractors, and Convergence

Calmed 3D Navier–Stokes Equations: Global Well-Posedness, Energy Identities, Global Attractors, and Convergence

EXISTENCE OF THE GLOBAL ATTRACTOR FOR A HYPERBOLIC PHASE FIELD SYSTEM OF CAGINALP TYPE WITH RELAXATION, GOVERNED BY A POLYNOMIAL GROWTH POTENTIAL OF DEGREE

Phase field systems have attracted the attention of many researchers in physics and mathematics for several years. They have a wide range of applications, including materials science, thermal welding, crystal formation, and others. In such phenomena, perturbations can occur. The industrial world seeks to understand the behavior of the perturbed system while seeking better control of the perturbation. Our goal in this paper is to study the asymptotic behavior of the solution, proving the existence of the global attractor for perturbed phase field systems with a regular potential and polynomial growth typically of degree .

Approximation of the long-time dynamics of the dynamical system generated by the Ericksen–Leslie equations

In this article we study the long-time dynamics of the dynamical system generated by the Ericksen–Leslie model. More precisely, we discretize the Ericksen–Leslie equations in time using the implicit Euler scheme, and with the aid of the discrete Gronwall lemmas we prove that the scheme is uniformly bounded. Moreover, using the theory of the multi-valued attractors we prove in a particular case the convergence of the global attractors generated by the numerical scheme to the global attractor of the continuous system as the time-step approaches zero.

The assembly and dynamics of ecological communities in an ever‐changing world

AbstractAlternative perspectives on the maintenance of biodiversity and the assembly of ecological communities suggest that both processes cannot be investigated simultaneously. In this concept and synthesis, we challenge this view by presenting major theoretical advances in structural stability and permanence theory. These advances, which provide complementary views, allow studying the short‐ and long‐term dynamics of ecological communities as changes in species richness, composition, and abundance. Here, the global attractor, technically named informational structure (IS), is the central element to construct from information of species' intrinsic growth rates and their strength and sign of interactions. The global attractor has four main properties: (1) It contains all the limits of what is feasible and unfeasible of the dynamical behavior of an ecological system, therefore, (2) it provides a thorough characterization of all combinations of species' richness and composition in which species can coexist (i.e., feasible and stable equilibrium), (3) as well as all connections (paths) of assembly between coexisting communities. Importantly, (4) such topology of coexisting communities and their connections changes when environmental (abiotic and biotic) variation affects the ability of species to grow and interact with others. Overall, these four properties allow switching from a traditional evaluation of species coexistence at equilibrium to a much more realistic nonequilibrium perspective where changes in the structure of the global attractor underlie the transient ecological dynamics. Several fields in ecology can benefit from the study of an IS. For instance, it can serve to evaluate community responses after the end of a perturbation, to design restoration trajectories, to study the consequences of biological invasions on the persistence of native species within communities, or to assess ecosystem health status. We illustrate this latter possibility with empirical observations of 7 years in Mediterranean annual grasslands. We document that extremely wet or dry years generate ISs supporting few coexisting communities and few assembly paths. The remaining communities distinguish winners from losers of ongoing climate change and indicate the limits to future community assembly opportunities. A fully tractable operational framework is readily available to understand and predict the assembly and dynamics of ecological communities in an ever‐changing world.

Asymptotic analysis of the 2D and 3D convective Brinkman–Forchheimer equations in unbounded domains

In this work, we carry out the asymptotic analysis of two- and three-dimensional convective Brinkman–Forchheimer (CBF) equations, which characterize the motion of incompressible fluid flows in a saturated porous medium. We establish the existence of a global attractor in both bounded (using compact embedding) and unbounded Poincaré domains (using asymptotic compactness property). In Poincaré domains, the estimates for Hausdorff as well as fractal dimensions of the global attractors are also obtained. We then show an upper semicontinuity of global attractors with respect to domains for CBF equations. We consider an expanding sequence of simply connected, bounded and smooth subdomains [Formula: see text] of the Poincaré domain [Formula: see text] such that [Formula: see text] as [Formula: see text]. If [Formula: see text] and [Formula: see text] are the global attractors of CBF equations corresponding to [Formula: see text] and [Formula: see text], respectively, then we show that for large enough [Formula: see text], the global attractor [Formula: see text] enters into any neighborhood [Formula: see text] of [Formula: see text]. The presence of the Darcy term in CBF equations helps us to obtain the above mentioned results in general unbounded domains also. Finally, we discuss the quasi-stability property of the semigroup associated with CBF equations in bounded domains and establish the existence of finite fractal dimensional global as well as exponential attractors.

Birkhoff attractors of dissipative billiards

Abstract We study the dynamics of dissipative billiard maps within planar convex domains. Such maps have a global attractor. We are interested in the topological and dynamical complexity of the attractor, in terms both of the geometry of the billiard table and of the strength of the dissipation. We focus on the study of an invariant subset of the attractor, the so-called Birkhoff attractor. On the one hand, we show that for a generic convex table with ‘pinched’ curvature, the Birkhoff attractor is a normally contracted manifold when the dissipation is strong. On the other hand, for a mild dissipation, we prove that, generically, the Birkhoff attractor is complicated, both from the topological and the dynamical points of view.

Global well-posedness and dynamics of spatial diffusion HIV model with CTLs response and chemotaxis

In this paper, we study the global well-posedness and global dynamics of a reaction–diffusion HIV infection model with the chemotactic movement of CTLs (Cytotoxic T lymphocytes). We first show the global existence and uniform boundedness for solutions of the system with general functional incidences. Then, for the model with bilinear incidence rate, we discuss the existence conditions of the three equilibria (infection-free, chemokines-extinct, chemokines-acute equilibria) of the model and obtain the conclusion of the local asymptotic stability of these equilibria by analyzing the linearized system at these equilibria. Moreover, by constructing reasonable Lyapunov functionals, we investigate the global stability and attractivity of the equilibria. Applying the Lp−Lq estimate, Young’s inequality, Gagiardo-Nirenberg inequality and the parabolic regularity theorem, we also discuss the convergence rates of the equilibria. Finally, some numerical simulations are conducted to verify the theoretical results.

Global and exponential attractors for viscoelastic Kirchhoff plate equation with memory and time delay

In this paper, we consider the viscoelastic Kirchhoff plate equation with memory and time delay. Under appropriate conditions on the real numbers and , we prove the existence of a compact global attractor with finite fractal dimension through a stabilizability estimate. Additionally, we demonstrate the existence of a fractal exponential attractor. To the best of our knowledge, these findings are novel for for this system.