Asymptotic Stability Of Equilibrium Point Research Articles

A proximal neurodynamic model for a system of non-linear inverse mixed variational inequalities

In this article, we introduce a system of non-linear inverse mixed variational inequalities (SNIMVIs). We propose a proximal neurodynamic model (PNDM) for solving SNIMVIs, leveraging proximal mappings. The uniqueness of the continuous solution for the PNDM is proved by assuming Lipschitz continuity. Moreover, we establish the global asymptotic stability of equilibrium points of the PNDM, contingent upon Lipschitz continuity and strong monotonicity. Additionally, an iterative algorithm involving proximal mappings for solving the SNIMVIs is presented. Finally, we provide illustrative examples to support our main findings. Furthermore, we provide an example where the SNIMVIs violate the strong monotonicity condition and exhibit the divergence nature of the trajectories of the corresponding PNDM.

Analyzing dynamics and stability of single delay differential equations for the dengue epidemic model

This paper introduces a mathematical model that simulates the transmission of the dengue virus in a population over time. The model takes into account aspects such as delays in transmission, the impact of inhibitory effects, the loss of immunity, and the presence of partial immunity. The model has been verified to ensure the positivity and boundedness. The basic reproduction number R0 of the model is derived using the advanced next-generation matrix approach. An analysis is conducted on the stability criteria of the model, and equilibrium points are investigated. Under appropriate circumstances, it was shown that there is local stability in both the virus-free equilibrium and the endemic equilibrium points when there is a delay. Analyzing the global asymptotic stability of equilibrium points is done by using the appropriate Lyapunov function. In addition, the model exhibits a backward bifurcation, in which the virus-free equilibrium coexists with a stable endemic equilibrium. By using a sensitivity analysis technique, it has been shown that some factors have a substantial influence on the behavior of the model. The research adeptly elucidates the ramifications of its results by effortlessly validating theoretical concepts with numerical examples and simulations. Furthermore, our research revealed that augmenting the rate of inhibition on infected vectors and people leads to a reduction in the equilibrium point, suggesting the presence of an endemic state.

Mathematical modelling and projection of Buruli ulcer transmission dynamics using classical and fractional derivatives: A case study of Cameroon

The goal of this work is to derive and study a new model which traduces the transmission dynamics of the Buruli ulcer (BU), in which we replace mass action incidences with standard incidences, consider latent period, and by using both integer and fractional derivatives. We first present reported data of Buruli in Cameroon between 01/01/2001 and 31/12/2014, and the statistical model that approximates real data as closely as possible, and thus make a forecasting in the next sixteen years. After that, we formulate a Susceptible humans-Exposed humans-Infectious humans-Recovered humans-Susceptible humans; Susceptible vectors-Infectious vectors (SEIRS-SI) type compartmental model of Buruli Ulcer using integer derivative. We compute the basic reproduction number denoted by R0 and prove the asymptotic stability of the Buruli-free equilibrium whenever R0<1. Then, for R0>1, we prove the existence of an unique endemic equilibrium point and its global stability using the general theory of Lyapunov. We perform parameter estimation to calibrate our model using real data from Cameroon, while sensitivity analysis is conducted to determine important parameters in the BU dynamics. From this model calibration, we obtain R0=2.0843 which is greater than one and implies that BU ulcer is endemic in the country. To determine whether if or not the BU admits one or multiple waves, we compute the strength number denoted by A0. We find that A0≈17>0 which means that Bu admits multiple epidemic waves. We then replace integer derivative with the Caputo fractional derivative. Asymptotic stability of equilibrium points are also performed for the fractional model. This follows by the proof of existence and the uniqueness of solutions of the fractional model. We then construct a numerical scheme based on the Adams–Bashforth–Moulton (ABM) Method. Theoretical results are validated from numerical simulations. These last also permits to evaluate the impact of varying fractional order α in the BU dynamics. This permits to conclude that for α<1, reported cases are closer to model prediction.

Population of entities with three individual states and asymmetric interactions

In various applications in life and social sciences the agents of a complex system, which can be individuals of a sub–populations, can have one of the few inner states or can choose one of few strategies. Their states are effects of interactions with other agents. Behavior is described by a kinetic–like a nonlinear equation. In the present paper, we study the behavior of solutions in the case of three possible states and show that in some cases a kind of self–organization occurs but in others, a periodic behavior characterizes the system. We can observe a wide variety of dynamics that can relate to the behavior of real systems. The model contains a natural interaction intensity parameter γ≥1. Case γ>1 leads to “very” nonlinear structures. We prove that the system has no non–constant periodic solutions. We propose conditions guaranteeing the asymptotic stability of equilibrium points on the boundary that corresponds to the asymptotic extinction of two states. Moreover, conditions for the uniqueness and instability of an inner equilibrium point, corresponding to an asymptotic presence of all states, are formulated. The case of γ=1 with asymmetric interaction rate is studied as well. Possible complex behavior of solutions can reflect the possible complex performance of systems with asymmetric interactions — typical e.g. in Economy and some applications in Biology.

On Asymptotic Properties of Stochastic Neutral-Type Inertial Neural Networks with Mixed Delays

This article studies the stability problem of a class of stochastic neutral-type inertial delay neural networks. By introducing appropriate variable transformations, the second-order differential system is transformed into a first-order differential system. Using homeomorphism mapping, standard stochastic analyzing technology, the Lyapunov functional method and the properties of a neutral operator, we establish new sufficient criteria for the unique existence and stochastically globally asymptotic stability of equilibrium points. An example is also provided, to show the validity of the established results. From our results, we find that, under appropriate conditions, random disturbances have no significant impact on the existence, stability, and symmetry of network systems.

Fractional dynamics of a Chikungunya transmission model

In this paper, we introduce a Caputo-sense fractional derivative to an existing model for the Chikungunya transmission dynamics, replacing integer derivative. After reviewing previous work on the integer order model, we suggest a change whereby the integer derivative is replaced by the Caputo derivative operator, and then we obtain results about the asymptotic stability of equilibrium points in this new fractional model. We prove the existence and uniqueness of the solution as well as the global stability of the fractional system. We then construct a numerical scheme for the fractional model, and use parameter values from the Chikungunya epidemic in Chad to perform numerical simulations. This allows us to validate our theoretical results and compare the dynamics behaviour of both models. We find from numerical simulations, that for the fractional-order parameter η in the range (0.85;1), daily detected cases are closer to those the model predict. Thus in the case of Chikungunya epidemic in Chad, the model with fractional derivative produces better results than which with integer derivative.

ОЦЕНКИ РЕШЕНИЙ ДЛЯ ОДНОЙ МОДЕЛИ ДИНАМИКИ ПОПУЛЯЦИЙ С ЗАПАЗДЫВАНИЕМ

We consider a model of the single population dynamics described by a delay differential equation. The asymptotic behavior of solutions for this model is studied in cases of asymptotic stability of equilibrium points corresponding to the complete extinction of the population and to the constant positive population size. In each case, Lyapunov–Krasovskii functionals are constructed, with the help of which estimates characterizing the rate of extinction of the population and the rate of stabilization of the population to a constant value are obtained.

Stability and Hopf Bifurcation Analysis of a Fractional-order Leslie-Gower Prey-predator-parasite System with Delay

A fractional-order Leslie-Gower prey-predator-parasite system with delay is proposed in this article. The existence and uniqueness of the solutions, as well as their non-negativity and boundedness, are studied. Based on the characteristic equations and the conditions of stability and Hopf bifurcation, the local asymptotic stability of each equilibrium point and Hopf bifurcation of interior equilibrium point are investigated. Moreover, a Lyapunov function is constructed to prove the global asymptotic stability of the infection-free equilibrium point. Lastly, numerical examples are studied to verify the validity of the obtained newly results

Stability and Bifurcation Analysis of a Fractional-Order Food Chain Model with Two Time Delays

In this study, the stability and bifurcation problems of a fractional food chain system with two kinds of delays are studied. Firstly, the nonnegative, bounded, and unique properties of the solutions of the system are proved. The asymptotic stability of the equilibrium points of the system is discussed. Furthermore, the global asymptotic stability of the positive equilibrium point is deduced by using Lyapunov function method. Secondly, the system takes two kinds of time delays as bifurcation parameters and calculates the critical values of Hopf bifurcation accurately. The results show that Hopf bifurcation can advance with increasing fractional order and another delay. In conclusion, numerical simulation verifies and illustrates the theoretical results.

Asymptotic stability for quaternion-valued BAM neural networks via a contradictory method and two Lyapunov functionals

&lt;abstract&gt;&lt;p&gt;We explore the existence and asymptotic stability of equilibrium point for a class of quaternion-valued BAM neural networks with time-varying delays. Firstly, by employing Homeomorphism theorem and a contradictory method with novel analysis skills, a criterion ensuring the existence of equilibrium point of the considered quaternion-valued BAM neural networks is acquired. Secondly, by constructing two Lyapunov functionals, a criterion assuring the global asymptotic stability of equilibrium point for above discussed quaternion-valued BAM is presented. Applying a contradictory method to study the equilibrium point and applying two Lyapunov functionals to study stability of equilibrium point are completely new methods.&lt;/p&gt;&lt;/abstract&gt;

Compositional Analysis of Interconnected Systems Using Delta Dissipativity

This letter extends the notion of delta dissipativity, originally introduced in a game theoretic context, to general interconnections of dynamical systems. The main contribution of this letter is a compositionality result that presents conditions under which a large-scale interconnection of delta dissipative systems is delta dissipative. We adapt this result to also analyze stability and asymptotic stability of equilibrium points for the interconnection. Additionally, we formulate a sum-of-squares program for verifying delta dissipativity of a (polynomial) system. The results are illustrated with examples.

Novel criteria for robust stability of Cohen-Grossberg neural networks with multiple time delays

&lt;p style='text-indent:20px;'&gt;This research paper deals with the investigation of global robust stability results for Cohen-Grossberg neural networks involving the multiple constant time delays. The activation functions in this neural network model are supposed to be in the set of non-decreasing slope-bounded nonlinear functions and the uncertainties in the constant network parameters are considered to have bounded upper norms. By employing a proper positive definite Lyapunov-type functional and using homeomorphism mapping theory, we propose some novel sets of novel conditions that assure both existence, uniqueness and global robust asymptotic stability of equilibrium points of this nonlinear Cohen-Grossberg-type neural network model involving the multiple time delays. The derived robustly stable conditions mainly rely on examining some proper relationships that are imposed on constant valued interconnection matrices of this delayed neural network. These stability conditions can be certainly verified by employing various simple and useful properties of real interval matrices. Some comparisons are made to address the key advantages of these novel criteria over previously reported corresponding results. An instructive example is also examined to observe novelty of these proposed criteria.&lt;/p&gt;

Novel criteria for global robust stability of dynamical neural networks with multiple time delays

This research article considers the problem regarding global robust asymptotic stability of the general type of dynamical neural networks involving multiple constant time delays. Some new sufficient criteria are proposed for the existence, uniqueness and global asymptotic stability of the equilibrium point of this neural network model whose network parameters possess uncertainties. This paper will first address the existence and uniqueness problem for equilibrium points by utilizing the Homomorphic transformation theory. Secondly, by exploiting a novel Lyapunov functional candidate, the sufficient conditions for asymptotic stability of equilibrium points of this class of delayed neural networks will be established. The derived robust stability conditions are expressed independently of the constant time delay parameters and define some novel relationships among network parameters of the considered neural network. Thus, the applicability and validity of the obtained robust stability conditions for delayed-type neural networks can be easily tested. The comprehensive comparisons among the results of the current article and some of previously derived corresponding results will also be made by giving an illustrative numerical example.

Effect of quarantine on transmission dynamics of Ebola virus epidemic: a mathematical analysis

In this paper, a nonlinear model called susceptible–exposed–infected–quarantined–recovered is developed to study the transmission of Ebola virus disease (EVD). In the proposed model, an additional class of quarantined humans is incorporated to investigate the impact of quarantine strategy for exposed population. A comprehensive mathematical analysis of this model is carried out to understand the dynamical behavior of EVD. Equilibrium points $$F^{0}$$ , $$F^{1}$$ and the threshold parameter $${\mathcal {R}}_{0}$$ of the model are evaluated. An analytic stability analysis of equilibrium points with the help of $${\mathcal {R}}_{0}$$ is performed. It is observed that $$F^{0}$$ is both locally and globally asymptotically stable when $${\mathcal {R}}_{0}$$ is strictly less than unity which means that there will be no epidemic. Moreover, $$F^{0}$$ is not stable and $$F^{1}$$ is both locally and globally asymptotically stable when $${\mathcal {R}}_{0}$$ is strictly greater than unity which indicates a uniform spread of disease among individuals. Global stability of both equilibria is established by employing theory of Lyapunov functions. To validate theoretical results thus obtained, the system of ODEs is solved by employing three well-known different numerical methods such as Euler method, Runge–Kutta method of order 4 (RK4), and the nonstandard finite difference (NSFD) method. It is worth mentioning that the Euler and RK4 numerical schemes converge conditionally, whereas the proposed NSFD scheme converges unconditionally and is dynamically consistent with the continuous model. A quantitative analysis of the proposed model at endemic point $$F^{1}$$ for different quarantine levels is also presented. We have studied the effect of different quarantine coverage levels on threshold parameter $${\mathcal {R}}_{0}$$ to draw our conclusions. Numerical results drawn using MATLAB validate our claim that EVD could be eradicated faster if quarantine measures with proper awareness at various coverage levels are adopted. Global asymptotical stability of equilibrium points is shown by 3D plots. Finally, the joint variability of populations is executed to assess the impact of quarantine measures on the transmission dynamics of a disease.

Lyapunov stability analysis of a mass–spring system subject to friction

Abstract This paper deals with the stability analysis of a mass–spring system subject to friction using Lyapunov-based arguments. As the described system presents a stick–slip phenomenon, the mass may then periodically stick to the ground. The objective consists of developing numerically tractable conditions ensuring the global asymptotic stability of the unique equilibrium point. The proposed approach merges two intermediate results: The first one relies on the characterization of an attractor around the origin, to which converges the closed-loop trajectories. The second result assesses the regional asymptotic stability of the equilibrium point by estimating its basin of attraction. The main result relies on conditions allowing to ensure that the attractor issued from the first result is included in the basin of attraction of the origin computed from the second result. An illustrative example draws the interest of the approach.

On Stability Analysis of Higher-Order Rational Difference Equation

In this paper, we study the equilibrium points, local asymptotic stability of equilibrium points, global behavior of equilibrium points, boundedness and periodicity of the rational recursive sequence wn+1=wn−pα+βwn/γwn+δwn−r, where γwn≠−δwn−r for r∈0,∞, α, β, γ, δ∈0,∞, and r&gt;p≥0. With initial values w−p,w−p+1,…,w−r,w−r+1,…,w−1, and w0 are positive real numbers. Some numerical examples are given to verify our theoretical results.

Global asymptotic stability for discrete-time Cohen-Grossberg neural networks with delays by combining graph theoretic approach with Homeomorphism concept

ABSTRACT In this paper, global asymptotic stability for a class of discrete-time Cohen-Grossberg Neural Networks with finite and infinite delays is investigated. By combining graph theoretic approach with Homeomorphism concept as well as Lyapunov functional method, two new sufficient conditions ensuring the global asymptotic stability of equilibrium point for above neural networks are established. Combining graph theoretic approach with Homeomorphism concept studies the equilibrium point of neural networks is a novel approach.

Dynamic Behaviors of a Class of High-Order Fuzzy Difference Equations

The purpose of this paper is to give the conditions for the existence and uniqueness of positive solutions and the asymptotic stability of equilibrium points for the following high-order fuzzy difference equation:xn+1=Axn−1xn−2/B+∑i=3kCixn−i n=0,1,2,…,wherexnis the sequence of positive fuzzy numbers and the parametersA,B,C3,C4,…,Ckand initial conditionsx0,x−1,x−2,x−ii=3,4,…,kare positive fuzzy numbers. Besides, some numerical examples describing the fuzzy difference equation are given to illustrate the theoretical results.

Modeling pyrethroids repellency and its role on the bifurcation analysis for a bed net malaria model

• Bed net lethal and repellent effects are incorporated in a dynamical malaria model and the disease contact rates are explicitly modeled by highly nonlinear functions of their rates. • The complete asymptotic behavior of the model is established, using Lyapunov-LaSalle techniques and geometric approach. • An in-depth bifurcation analysis is performed, with all the equilibrium points topologically classified and the existence of trans-critical forward and backward bifurcations established. • Bed net repellency and disease-induced mortality are solely or jointly the causes of backward bifurcation, which impedes the classical requirement that, bringing the basic reproduction number below unity is sufficient to control malaria. • Bed nets with high lethal effect is better that bed nets with high repellency effect.Both bed net repellency and lethal effects decrease disease contact rates, and lethal effect rate does it faster than repellency effect. • Bed net repellency effect regulates the disease burden and its negligence underestimates that burden. We develop and analyze a simple mathematical model for malaria transmission where pyrethroids treated nets (PTNs) are used for control purposes, and in which knock-down/lethal, excito-repellent/deterrent effects are incorporated. We explicitly describe the contact rates between mosquitoes and humans by nonlinear functions of bed net usage and repellency rates. Using center manifold theory, we show that our model exhibits, saddle-node, transcritical forward and backward bifurcation when the reproduction number R 0 crosses one. Our model reveals that repellency effect plays an important role for the existence of both the endemic equilibrium points and the occurrence of backward bifurcation and a threshold repellency rate is calculated. The epidemiological implication of the backward bifurcation is that, reducing R 0 below one alone is not enough to eliminate malaria. We establish that, increasing either the rate of bed nets usage (i.e community protection) or their repellent effect (i.e personal protection) or the combination of both, decreases the contact rates between humans and mosquitoes. As a result, the disease burden metric R 0 is reduced. The global asymptotic stability of equilibrium points are proven using the geometric approach and Lyapunov-LaSalle techniques. Furthermore, we show that neglecting repellency underestimates the basic reproduction number and hinders the control of malaria. The disease free equilibrium is shown to be a saddle-node of co-dimension 1 when R 0 = 1 . We also observe that R 0 is mostly influenced by the bed net coverage rate, repellent effect of pyrethroids and the probability that mosquitoes target human hosts. Our results confirm that PTNs usage is an efficient control strategy to mitigate the malaria ability to spread, and suggest that the utilization of PTNs with high lethal rate, but low repellency rate is better than the use of those with high repellency and low lethal rates.

Multiple Lagrange stability and Lyapunov asymptotical stability of delayed fractional-order Cohen–Grossberg neural networks**Project supported by the Natural Science Foundation of Zhejiang Province of China (Grant Nos. LY18F030023, LY17F030016, LQ18F030015, and LY18F020028) and the National Natural Science Foundation of China (Grant Nos. 61503338, 61773348, and 61972354).

This paper addresses the coexistence and local stability of multiple equilibrium points for fractional-order Cohen–Grossberg neural networks (FOCGNNs) with time delays. Based on Brouwer’s fixed point theorem, sufficient conditions are established to ensure the existence of equilibrium points for FOCGNNs. Through the use of Hardy inequality, fractional Halanay inequality, and Lyapunov theory, some criteria are established to ensure the local Lagrange stability and the local Lyapunov asymptotical stability of equilibrium points for FOCGNNs. The obtained results encompass those of integer-order Hopfield neural networks with or without delay as special cases. The activation functions are nonlinear and nonmonotonic. There could be many corner points in this general class of activation functions. The structure of activation functions makes FOCGNNs could have a lot of stable equilibrium points. Coexistence of multiple stable equilibrium points is necessary when neural networks come to pattern recognition and associative memories. Finally, two numerical examples are provided to illustrate the effectiveness of the obtained results.