Global Attractor Research Articles

New discussion on global existence and attractivity of mild solutions for nonautonomous integrodifferential equations with state-dependent delay

New discussion on global existence and attractivity of mild solutions for nonautonomous integrodifferential equations with state-dependent delay

Statistical performance of local attractor dimension estimators in non-Axiom A dynamical systems.

We investigate various estimators based on extreme value theory (EVT) for determining the local fractal dimension of chaotic dynamical systems. In the limit of an infinitely long time series of an ergodic system, the average of the local fractal dimension is the system's global attractor dimension. The latter is an important quantity that relates to the number of effective degrees of freedom of the underlying dynamical system, and its estimation has been a central topic in the dynamical systems literature since the 1980s. In this work, we propose a framework that combines phase space recurrence analysis with EVT to estimate the local fractal dimension around a particular state of interest. While the EVT framework allows for the analysis of high-dimensional complex systems, such as the Earth's climate, its effectiveness depends on robust statistical parameter estimation for the assumed extreme value distribution. In this study, we conduct a critical review of several EVT-based local fractal dimension estimators, analyzing and comparing their performance across a range of systems. Our results offer valuable insights for researchers employing the EVT-based estimates of the local fractal dimension, aiding in the selection of an appropriate estimator for their specific applications.

Structural stability and rate of convergence of global attractors

In this paper, we prove that a Lipschitz perturbed map of a gradient Morse–Smale diffeomorphism T on Rm has the Lipschitz shadowing property on a neighborhood of the global attractor AT, and apply the result to get the structural stability and the rate of convergence of global attractors of Chafee–Infante equations under Lipschitz perturbations of the domain and equation.

Further Studies on the Dynamics of a Lotka–Volterra Competitor–Competitor–Mutualist System with Time-Varying Delays

In this paper, a Lotka–Volterra (L-V) competitor–competitor–mutualist system with time-varying delays is studied. Some dynamical behaviors of the considered system are investigated. Firstly, we obtain the boundedness, permanence and periodic solution of the system using the comparison principle of differential equations and inequality estimation method. Then, the global attractiveness of the system is analyzed by multiple Lyapunov functionals. Meanwhile, the existence and global attractivity of positive periodic solutions is derived. In the third section, in order to validate the practicability and feasibility of the obtained theoretical results, we conducted numerical simulations using MATLAB function ddesd. Finally, the fourth section is where conclusions are drawn.

The existence conditions for global exponential attractor of non-autonomous evolution equations and applications

Global exponential attractor theory is an effective tool for studying the dynamic of dissipative evolution equations. However, the existence of global exponential attractor for non-autonomous dynamic system is not only a difficult problem, but also an unsolved problem. In this paper, according to the structure of evolution equations, we propose two equivalent conditions for the existence of global exponential attractor of non-autonomous evolution equations. Using these two equivalent conditions, we prove the existence of bounded absorbing set, the existence of global exponentially attracting set for non-autonomous dissipative evolution equations, and study the finite fractal dimension estimation for the global exponentially attracting set of non-autonomous dissipative evolution equations. Finally, as applications, the existence of global exponential attractor is proved for non-autonomous semilinear parabolic equation and the epidemic model. Our results also provide a convenient and effective tool to study the global exponential attractor.

Long-time dynamics of a problem of strain gradient porous elastic theory with nonlinear damping and source terms

Of concern is a problem of strain gradient porous elastic theory with nonlinear damping terms, whose constitutive equations contain higher-order derivatives of the displacement in the basic postulates. The paper is based on the theory of ‘consistency’ due to Aouadi et al. [J. Therm. Stress. 43(2)(2020), 191–209] and Ieşan [American Institute of Physics, Conference Proceedings, 1329 (2011), 130–149], and contains four results. We firstly show that the system is global well posed by using maximal monotone operator. The second main result is the existence of global attractors which is proved by the method developed by Chueshov and Lasiecka [Long-time behavior of second order evolution equations with nonlinear damping. Mem. Amer. Math. Soc. vol. 195, no. 912, Providence, 2008; Von Karman evolution equations: well-posedness and long-time dynamics. Springer Monographs in Mathematics, Springer, New York, 2010]. By showing the system is gradient and asymptotically smooth, we establish the existence of global attractors with finite fractal dimension via a stabilizability inequality. Then we study the continuity of global attractors regarding the parameter in a residual dense set. The above results allow the damping terms with polynomial growth. Finally we discuss the exponential decay and global boundedness to the linear case of damping terms of the system. The assumption of equal-speed wave propagations is not needed for all of results obtained.

Comparative data-driven enhanced geothermal systems forecasting models: A case study of Qiabuqia field in China

Geothermal energy is gaining global attractiveness owing to its abundance and sustainable nature. An in-depth understanding of potential geothermal production provides the energy industry a possibility to diversify the supply portfolio. With the development of artificial intelligence, machine learning offers an efficient alternative to the conventional numerical simulation method in forecasting energy harvesting. However, a comprehensive comparison and an effective algorithm selection are absent from the machine learning applications in forecasting geothermal energy recovery. In this study, four machine learning algorithms based data-driven models are created to determine the optimal choice in predicting geothermal production, including K-Nearest Neighbors (KNN), Support Vector Machine (SVM), Extreme Gradient Boosting (XGBoost) and Artificial Neural Network (ANN). To investigate their application range, two different sizes of data groups are involved to train and test these models, and their performance is comprehensively compared. As the results show, the highest coefficient of determination R2 of 0.998 is demonstrated in the ANN models showing its promising predictive ability. Besides, the ANN is the most stable with the lowest performance variances between a training set and a validation set. In addition, the ANN is the most adaptable due to its minimal performance differences between different sizes of data groups. By jointly considering the prediction accuracy, stability and adaptability, the ANN is the best choice to substitute numerical simulation for predicting geothermal development. Importantly, the successful implementation of the proposed data-driven model requires 2700 times less computational time compared to numerical simulation, demonstrating a considerable improvement in the prediction efficiency. The results provide a beneficial reference for operators in conducting machine learning to simulate the development of the geothermal system studied, and can be effectively applied in other energy systems.

Global attractors of generic reaction diffusion equations under Lipschitz perturbations

In this paper we prove that generically the dynamical system induced by a reaction diffusion equation is Gromov-Hausdorff stable on its global attractor under Lipschitz perturbations of the domain and equation.

Global attractors for porous-elasticity system from second spectrum viewpoint

This paper is concerned with the global attractors for a new partially damped porous-elasticity system by taking a truncated version which is free of blow-up on second wave speed. We establish the global well-posedness of the system via Faedo–Galerkin method. By considering only one damping term acting on the volume fraction in the system we prove the existence of absorbing set for the solution semigroup regardless any relationship between coefficients of the system. Finally, by using Lyapunov and recent quasi-stability methods we prove the existence of smooth global attractors with finite fractal dimension.

Dynamics of a diffusive delayed viral infection model in a heterogeneous environment

This paper investigates the asymptotic analysis of spatially heterogeneous viral transmission, incorporating cell‐to‐cell transmission, virus nonlocal dispersal, and intracellular delay. Due to the noncompactness of the semiflow, we used the Kuratowski measure of noncompactness to demonstrate the existence of a global compact attractor. This noncompactness issue generates difficulties in calculating the basic reproduction number , which is the principal eigenvalue of the next‐generation operator. The threshold role of this number is determined, where we derived two different cases: (i) the global stability of the virus‐free steady state, which is globally stable for by the Lyapunov direct method, and (ii) the global stability of the virus steady state for . Indeed, the second case is demonstrated through several steps that include uniform persistence, the existence of a virus steady state, and the global stability of the virus steady state. The results are supported by different graphical representations with proper biological justifications.

On global dynamics of type-K competitive Kolmogorov differential systems

This paper deals with global asymptotic behaviour of the dynamics for N-dimensional type-K competitive Kolmogorov systems of differential equations defined in the first orthant. It is known that the backward dynamics of such systems is type-K monotone. Assuming the system is dissipative and the origin is a repeller, it is proved that there exists a compact invariant set Σ which separates the basin of repulsion of the origin and the basin of repulsion of infinity and attracts all the non-trivial orbits. There are two closed sets S H and S V , their restriction to the interior of the first orthant are -dimensional hypersurfaces, such that the asymptotic dynamics of the type-K system in the first orthant can be described by a system on either S H or S V : each trajectory in the interior of the first orthant is asymptotic to one in S H and one in S V . Geometric and asymptotic features of the global attractor Σ are investigated. It is proved that the partition holds such that and . Thus, Σ0 contains all the ω-limit sets for all interior trajectories of any type-K subsystems and the closure as a subset of Σ is invariant and the upper boundary of the basin of repulsion of the origin. This Σ has the same asymptotic feature as the modified carrying simplex for a competitive system: every nontrivial trajectory below Σ is asymptotic to one in Σ and the ω-limit set is in Σ for every other nontrivial trajectory.

Experiments on the Artificial Potential Field with Local Attractors for Mobile Robot Navigation

Obstacle avoidance is a challenging task in robot navigation, as it requires efficient and reliable methods to avoid collision and reach the desired goal. Artificial potential field methods are widely used for this purpose, as they are efficient, effective, and easy to implement. However, they are limited by the use of only one global attractor at the goal. This paper introduces and evaluates experimentally a novel technique that enhances the artificial potential field method with local attractors. Local attractors can be positioned around the obstacle so as to guide the robot detouring through preferred regions. Thus, the side the robot will pass by can be determined in advance, making the collision-free path predictable. The technique is formulated by modelling local attractors as optimal inflections, i.e., regions that do not show local minima, which coexist with the potential field generated by the obstacle and the global attractor. The method is validated using a laboratory setup that employs a camera and markers to track the poses of the robot, the obstacle, and the target. A series of experiments are conducted to examine the effect of the local attractor under different test conditions, obtained by varying the obstacle pose, the attraction intensity, and the robot velocity. The experimental results demonstrate the effectiveness of the proposed technique and highlight the aspects that require further investigation for its improvement and application.

Exponential Convergence to Equilibrium for Coupled Systems of Nonlinear Degenerate Drift Diffusion Equations

We study the existence and long-time asymptotics of weak solutions to a system of two nonlinear drift-diffusion equations that has a gradient flow structure in the Wasserstein distance. The two equations are coupled through a cross-diffusion term that is scaled by a parameter . The nonlinearities and potentials are chosen such that in the decoupled system for , the evolution is metrically contractive, with a global rate . The coupling is a singular perturbation in the sense that for any , contractivity of the system is lost. Our main result is that for all sufficiently small , the global attraction to a unique steady state persists, with an exponential rate for some . The proof combines results from the theory of metric gradient flows with further variational methods and functional inequalities.

Poisson Stable Motions and Global Attractors of Symmetric Monotone Nonautonomous Dynamical Systems

This paper is dedicated to the study of the problem of existence of Poisson stable (Bohr/Levitan almost periodic, almost automorphic, almost recurrent, recurrent, pseudo-periodic, pseudo-recurrent and Poisson stable) motions of symmetric monotone non-autonomous dynamical systems (NDS). It is proved that every precompact motion of such system is asymptotically Poisson stable. We give also the description of the structure of compact global attractor for monotone NDS with symmetry. We establish the main results in the framework of general non-autonomous (cocycle) dynamical systems. We apply our general results to the study of the problem of existence of different classes of Poisson stable solutions and global attractors for a chemical reaction network and nonautonomous translation-invariant difference equations.

Global Attractor for a Coupled Wave and Plate Equation with Nonlocal Weak Damping on Riemannian Manifolds

Global Attractor for a Coupled Wave and Plate Equation with Nonlocal Weak Damping on Riemannian Manifolds

A relation-theoretic set-valued version of Prešić-Ćirić theorem and applications

In this paper, we establish a relation-theoretic set-valued version of the fixed point result of Ćirić and Prešić (Acta Math. Univ. Comen. LXXVI(2):143–147, 2007) on metric spaces endowed with an arbitrary binary relation. The results of this paper, generalize and unify the fixed point results of Ćirić and Prešić (Acta Math. Univ. Comen. LXXVI(2):143–147, 2007), Shukla and López (Quaest. Math. 45(3):1–16, 2019), and Shukla and Radenović (An. Ştiinţ. Univ. ‘Al.I. Cuza’ Iaşi, Mat. 63(2):339–350, 2017) in product spaces. Some examples are provided that justify and establish the importance of our results. As applications of our main result, we have established the existence of solutions to differential inclusion problems and the weak asymptotical stability and a global attractivity of the equilibrium point of a difference inclusion problem. The use of arbitrary binary relations in our results permits us to apply the results to the differential inclusion problems and difference inclusion problems with weaker assumptions than those used in the papers mentioned above.

A Class of Mutualistic Systems Having an Almost Periodic Global Attractor

A Class of Mutualistic Systems Having an Almost Periodic Global Attractor

Dynamics and profiles of a degenerated reaction–diffusion host-pathogen model with apparent and inapparent infection period

Inapparent infection plays an important role in the disease spread, which is an infection by a pathogen that causes few or no signs or symptoms of infection in the host. Many pathogens, including HIV, typhoid fever, and coronaviruses such as COVID-19 spread in their host populations through inapparent infection. In this paper, we formulated a degenerated reaction–diffusion host-pathogen model with multiple infection period. We split the infectious individuals into two distinct classes: apparent infectious individuals and inapparent infectious individuals, coming from exposed individuals with a ratio of (1−p) and p, respectively. Some preliminary results and threshold-type results are achieved by detailed mathematical analysis. We also investigate the asymptotic profiles of the positive steady state (PSS) when the diffusion rate of susceptible individuals approaches zero or infinity. When all parameters are all constants, the global attractivity of the constant endemic equilibrium is established. It is verified by numerical simulations that spatial heterogeneity of the transmission rates can enhance the intensity of an epidemic. Especially, the transmission rate of inapparent infectious individuals significantly increases the risk of disease transmission, compared to that of apparent infectious individuals and pathogens in the environment, and we should pay special attentions to how to regulate the inapparent infectious individuals for disease control and prevention, which is consistent with the result on the sensitive analysis to the transmission rates through the normalized forward sensitivity index. We also find that disinfection of the infected environment is an important way to prevent and eliminate the risk of environmental transmission.

Well‐posedness and dynamics for the von Karman equation with memory and nonlinear time‐varying delay

This paper is concerned with mathematical analysis for the viscoelastic von Karman equation with memory and time‐varying delay. Based on the existence and uniqueness of the strong solution established by Galerkin method, the finite dimensional global attractor for the system without time‐varying delay has been shown by using the quasi‐stability theory. The difficulties for our desired well‐posedness are the limiting of fourth‐order derivative terms and the von Karman bracket term, which need some delicate estimates to achieve. The key point for the dynamics is to obtain the quasi‐stability for gradient system, which needs the higher regularity of trajectory especially the von Karman bracket.

Bi-space Global Attractors for a Class of Second-Order Evolution Equations with Dispersive and Dissipative Terms in Locally Uniform Spaces

Bi-space Global Attractors for a Class of Second-Order Evolution Equations with Dispersive and Dissipative Terms in Locally Uniform Spaces