Global Attractor Research Articles

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

NmODE: neural memory ordinary differential equation

Brain neural networks are regarded as dynamical systems in neural science, in which memories are interpreted as attractors of the systems. Mathematically, ordinary differential equations (ODEs) can be utilized to describe dynamical systems. Any ODE that is employed to describe the dynamics of a neural network can be called a neuralODE. Inspired by rethinking the nonlinear representation ability of existing artificial neural networks together with the functions of columns in the neocortex, this paper proposes a theory of memory-based neuralODE, which is composed of two novel artificial neural network models: nmODE and epsilon-net, and two learning algorithms: nmLA and epsilon-LA. The nmODE (neural memory Ordinary Differential Equation) is designed with a special structure that separates learning neurons from memory neurons, making its dynamics clear. Given any external input, the nmODE possesses the global attractor property and is thus embedded with a memory mechanism. The nmODE establishes a nonlinear mapping from the external input to its associated attractor and does not have the problem of learning features homeomorphic to the input data space, as occurs frequently in most existing neuralODEs. The nmLA (neural memory Learning Algorithm) is developed by proposing an interesting three-dimensional inverse ODE (invODE) and has advantages in memory and parameter efficiency. The proposed epsilon-net is a discrete version of the nmODE, which is particularly feasible for digital computing. The proposed epsilon-LA (epsilon learning algorithm) requires no prior knowledge of the number of network layers. Both nmLA and epsilon-LA have no problem with gradient vanishing. Experimental results show that the proposed theory is comparable to state-of-the-art methods.

Permanence and Global Attractivity of a Non-autonomous Single Species System with Michaelis-Menten-Type Feedback Control

During the past decade, many scholars have investigated the dynamic behaviors of the ecosystem with Michaelis-Menten-type harvesting; however, most of them assume that the harvesting effort does not change with time. Such an assumption has its drawbacks. Generally speaking, the rate of increase in harvesting effort changes with the density of the species. Inspired by this, we put forth a novel form of single-population feedback control model, in which the feedback control variable is of the Michaelis-Menten-type. Sufficient conditions that ensure the permanence and global attractivity of the system are obtained.

Ramified continua as global attractors of C 1-smooth self-maps of a cylinder close to skew products

In the paper we consider -smooth self-maps of a cylinder close to -smooth skew products (and satisfying some additional conditions). We study such geometric property of the maps, as existence of -smooth invariant local lamination, and apply this geometric property to the proof of the geometric integrability of maps under consideration. Using obtained results we construct the example of the family of -smooth maps close to skew products so that each map from this family admits the global attractor, which is a one-dimensional ramified continuum with a complicated topological structure. The global attractor of every map from the family under consideration consists of arcs of two types. On the unique circle (which is the arc of first type) the map is mixing; on arcs of second type of different lengths homeomorphic to a closed interval (the family of such arcs has continuum cardinality) the map is not mixing. The topological structure of the global attractor and dynamical properties of trajectories on the attractor lead to the property of dense intermittency (in the complement to the attractor) of attraction sets of different ω-limit sets, the union of which coincides with the global attractor.

Dynamics in an n-Species Lotka–Volterra Cooperative System with Delays

We studied a class of generalized n-species non-autonomous cooperative Lotka–Volterra (L-V) systems with time delays. We obtained new criteria on the dynamic properties of the systems. First, we obtained the boundedness and permanence of the system using the inequality analysis technique and comparison method. Then, the existence of positive periodic solutions was investigated using the coincidence degree theory. The global attractivity of the system was obtained by constructing suitable Lyapunov functionals and utilizing Barbalat’s lemma. The existence and global attractivity of the periodic solutions were also obtained. Finally, we conducted two numerical simulations to validate the feasibility and practicability of our results.

Modeling and analysis of a diffusive cholera model with seasonally forced intrinsic incubation period and bacterial hyperinfectivity

In order to explore the impact of a seasonally forced intrinsic incubation period, bacterial hyperinfectivity, and environmental heterogeneity on the cholera spatiotemporal transmission dynamics, we construct a diffusive cholera model with a seasonal forced intrinsic incubation period and bacterial hyperinfectivity with non-flux boundary condition. The seasonally forced intrinsic incubation period is the period of time that pathogens spend in completing their development in the host, which extends the concept of the fixed latency period by introducing the seasonal impacts of the environment. We derive the basic reproduction number ℜ0 and investigate the threshold dynamics for exploring whether the model retains the core properties that are highly desirable in epidemic models, that is, the spread of cholera can be characterized by ℜ0. In a homogeneous case, we also establish the global attractivity of the unique constant endemic equilibrium by the technique of Lyapunov functional.

Quasi-stability and smooth global attractors for a elasticity-viscoporosity system with nonlinear damping and source of critical exponents

Quasi-stability and smooth global attractors for a elasticity-viscoporosity system with nonlinear damping and source of critical exponents

Dynamics of a diffusion dispersal viral epidemic model with age infection in a spatially heterogeneous environment with general nonlinear function

We propose a generalization of a model with age of infection in a heterogeneous environment. Firstly, we give the well‐posedness of the model and prove that the solutions are bounded and positive. The difficult mathematical issue in this research is that the model is partially degenerate, and the solution map is not compact. In addition, we construct a global attractor of a bounded set to establish the existence of total trajectory. Moreover, we define the principal eigenvalue associated with a principal eigenvalue problem to give a relation with the basic reproduction number and this value. By assuming that , then the infection‐free steady‐states is globally asymptotically stable. Furthermore, for and by using the persistence results, we prove the existence of endemic steady‐states , and by constructing an appropriate Lyapunov function, we show that is globally asymptotically stable. Lastly, we validate our theoretical analysis by giving some numerical graphics.

Analysis of a diffusive host–pathogen epidemic model with two‐stage mechanism in a spatially heterogeneous environment

Due to the spatial heterogeneity present in many aspects of disease transmission, the rate of transmission of infectious diseases, human birth/mortality rate, and the mobility ability of infected humans should be different in different geographical locations. On the other hand, infected humans may exhibit distinct differences in symptoms during the different stages of the disease transmission. This paper aims to study threshold dynamics of a reaction–diffusion host–pathogen model governed by two‐stage mechanism and no flux boundary condition. In the model, we use bilinear and saturated incidence mechanism to model the interactions between host populations and pathogens and assume that (i) the parameters are spatially dependent and (ii) infected individuals in the first stage can random walk, while in the second stage, the diffusion rates are neglected. By carrying out strict analysis, the paper establishes the threshold‐type results with the basic reproduction number. Specifically, in a homogeneous case that all parameters are constants, we establish the global attractivity of the endemic equilibrium. To investigate the effects of small or large diffusion rates of susceptible individuals on the disease transmission in a heterogeneous environment, we explore the asymptotic profile of positive steady state whenever it exists. Our numerical results validate the theoretical results and indicate the importance of the infected humans in the second stage and pathogens in the environment in disease transmission, which greatly contribute to disease spread in a bounded domain and should not be ignored.

Strongly compact strong trajectory attractors for evolutionary systems and their applications

We show that for any fixed accuracy and time length T, a finite number of T-time length pieces of the complete trajectories on the global attractor are capable of uniformly approximating all trajectories within the accuracy in the natural strong metric after sufficiently large time when the observed dissipative system is asymptotically compact. Moreover, we obtain the strong equicontinuity of all the complete trajectories on the global attractor. These results follow by proving the existence of a strongly compact strong trajectory attractor. The notion of a trajectory attractor was previously constructed for a family of auxiliary systems including the originally considered one without uniqueness. Recently, Cheskidov and the author developed a new framework called evolutionary system, with which a (weak) trajectory attractor can be actually defined for the original system. In this paper, the theory of trajectory attractors is further developed in the natural strong metric for our purpose. We then apply it to both the 2D and the 3D Navier–Stokes equations and a general nonautonomous reaction–diffusion system.

Existence and upper semicontinuity of random pullback attractors for 2D and 3D non-autonomous stochastic convective Brinkman-Forchheimer equations on whole space

In this work, we analyze the long time behavior of 2D as well as 3D convective Brinkman-Forchheimer (CBF) equations and its stochastic counterpart with non-autonomous deterministic forcing term in $\mathbb{R}^d$ $ (d=2, 3)$: $$ \frac{\partial\boldsymbol{u}}{\partial t}-\mu \Delta\boldsymbol{u} + (\boldsymbol{u}\cdot\nabla)\boldsymbol{u} + \alpha\boldsymbol{u} + \beta|\boldsymbol{u}|^{r-1} \boldsymbol{u} + \nabla p=\boldsymbol{f},\quad \nabla\cdot\boldsymbol{u}=0, $$ where $r\geq1$. We prove the existence of a unique global pullback attractor for non-autonomous CBF equations, for $d=2$ with $r\geq1$ and $d=3$ with $r > 3$ (for any $\beta,\mu > 0$), and $d=r=3$ with $2\beta\mu\geq1$. For the same cases, we show the existence of a unique random pullback attractor for non-autonomous stochastic CBF equations with linear multiplicative white noise. In addition, a remark on the existence of a unique weak pullback mean random attractor for the stochastic CBF equations is also provided. Finally, we establish the upper semicontinuity of the random pullback attractors, that is, the random pullback attractors converge towards the global pullback attractor when the noise intensity approaches to zero. Since we do not have compact Sobolev embeddings on unbounded domains, the pullback asymptotic compactness of the random dynamical system is proved by the method of energy equations given by Ball. For the case of Navier-Stokes equations defined on $\mathbb{R}^d$, such results are not available and the presence of Darcy term $\alpha\boldsymbol{u}$ helps us to establish the above mentioned results for CBF equations on whole space.