Liapunov Function Method Research Articles

Analysis of an epidemic model with latent infection

An infectious disease model with latent infection is studied, and the basic reproductio n number of the model is obtained When the basic reproducing number is less than 1, the mode l has only one disease-free equilibrium. When the basic reproducing number is greater than 1, t he disease-free equilibrium is unstable. The model also has a unique positive equilibrium. The global asymptotic stability of the two equilibria is analyzed by linearization method, Liapunov function method and geometric method.

Mathematical modeling and stochastic stability analysis of viscoelastic nanobeams using higher-order nonlocal strain gradient theory

This paper analyzes stochastic vibrations of a viscoelastic nanobeam under axial loadings. Based on the higher-order nonlocal strain gradient theory and the Liapunov functional method, bounds of the almost sure asymptotic stability of a nanobeam are obtained as a function of retardation time, variance of the stochastic force, higher-order and lower-order scale coefficients, strain gradient length scale, and intensity of the deterministic component of axial loading. Analytical results from this study are first compared with those obtained from the Monte Carlo simulation. Numerical calculations are performed for the Gaussian and harmonic non-white processes as models of axial forces.

Nonlinear orbital stability for planar vortex patches

In this paper, we prove nonlinear orbital stability for steady vortex patches that maximize the kinetic energy among isovortical rearrangements in a planar bounded domain. As a result, nonlinear stability for an isolated vortex patch is proved. The proof is based on conservation of energy and vorticity, which is an analogue of the classical Liapunov function method.

Nonlinear dynamics of avian influenza epidemic models

Avian influenza is a zoonotic disease caused by the transmission of the avian influenza A virus, such as H5N1 and H7N9, from birds to humans. The avian influenza A H5N1 virus has caused more than 500 human infections worldwide with nearly a 60% death rate since it was first reported in Hong Kong in 1997. The four outbreaks of the avian influenza A H7N9 in China from March 2013 to June 2016 have resulted in 580 human cases including 202 deaths with a death rate of nearly 35%. In this paper, we construct two avian influenza bird-to-human transmission models with different growth laws of the avian population, one with logistic growth and the other with Allee effect, and analyze their dynamical behavior. We obtain a threshold value for the prevalence of avian influenza and investigate the local or global asymptotical stability of each equilibrium of these systems by using linear analysis technique or combining Liapunov function method and LaSalle’s invariance principle, respectively. Moreover, we give necessary and sufficient conditions for the occurrence of periodic solutions in the avian influenza system with Allee effect of the avian population. Numerical simulations are also presented to illustrate the theoretical results.

On a periodic predator-prey system with nonlinear diffusion and delays

In this paper, we consider a two species Lotka–Volterra type predator-prey system with nonlinear diffusion and distributed time delays. Based on the continuation theorem of the coincidence degree theory and the Liapunov function method, some new sufficient conditions on the existence and global attractivity of positive periodic solutions are established.

Co-Existence of Local Limit Cycles from Degenerate and Weak Foci in Cubic Systems

In this paper, we investigate the existence of local limit cycles obtained by perturbing degenerate and weak foci of two-dimensional cubic systems of differential equations. In particular, we consider a specific class of such systems where the origin is a degenerate focus. By utilizing a Liapunov function method and the stability results that follow, we first determine constraints on the system to maximize the number of local limit cycles that can be obtained by perturbing the degenerate focus at the origin. Once this is established, we add on the additional assumption that the system has a weak focus at , where , and determine conditions to maximize the number of additional local limit cycles that can be obtained near this fixed point. We will ultimately achieve an example of a cubic system with three local limit cycles about the degenerate focus and one local limit cycle about the weak focus.

Asymptotic behavior of solutions for the time-delayed equations of Benjamin-Bona-Mahony’s type

In this paper, we investigate the asymptotic behavior of the solutions for the equations of Benjamin-Bona-Mahony’s type with a time delay. We prove the global existence of solutions and energy decay. By using the Liapunov function method, we shall show that the solution is exponentially decay if the delay parameter τ is sufficiently small.

Existence and global exponential stability of anti-periodic solutions of high-order bidirectional associative memory (BAM) networks with time-varying delays on time scales

In this paper, high-order bidirectional associative memory (BAM) networks with time-varying delays on time scales are investigated. With the help of the time scale calculus theory, differential inequality theory, some analysis skills and the Liapunov functional method, a set of sufficient conditions ensuring the existence and exponential stability of anti-periodic solutions for the high-order BAM neural network with time-varying delays are obtained. Our results are new and complement some previously known ones. Moreover, a numerical example is presented to verify the theoretical results.

Asymptotic Behavior of Solutions for the Time-Delayed Kuramoto-Sivashinsky Equation

In this paper, we investigate the asymptotic behavior of the solutions for the Kuramoto-Sivashinsky equation with a time delay. We prove the global existence of solutions and energy decay. By using the Liapunov function method, we shall show that the solution is exponentially decay if the delay parameter \tau is sufficiently small.

Dynamical Behavior for a Class of Delayed Competitive–Mutualism Systems

In this paper, a class of nonautonomous three-species Lotka–Volterra competitive–mutualism systems with time delays is discussed. Some sufficient conditions on the permanence of species and the global attractivity of the system are established by using the Liapunov function method and the method given in Nakata and Muroya (Nonlinear Anal 11:528–534, 2010).

Global Analysis of Beddington-DeAngelis Type Chemostat Model with Nutrient Recycling and Impulsive Input

In this paper, a Beddington-DeAngelis type chemostat model with nutrient recycling and impulsive input is considered. Except using Floquet theorem, introducing a new method combining with comparison theorem of impulse differential equation and by using the Liapunov function method, the sufficient and necessary conditions on the permanence and extinction of the microorganism are obtained. Two examples are given in the last section to verify our mathematical results. The numerical analysis shows that if only the system is permanent, then it also is globally attractive.

On blowup phenomena of solutions to the Euler equations for Chaplygin gases

In this paper, we are concerned with the blowup phenomena of solutions to the isentropic Euler equations for Chaplygin gases. By the method of phase portrait and Liapunov functions, we first obtain the blowup results of the spherically symmetric solutions to the equations in the multi-dimensional case. For one-dimensional Chaplygin gases, some solutions globally in time and blowup phenomena are derived simultaneously by the constructing method.

Design of second order neural networks as dynamical control systems that aim to minimize nonconvex scalar functions

This paper presents a unified way to design neural networks characterized as second order ordinary differential equations (ODEs) that aim to find the global minimum of nonconvex scalar functions. These neural networks, alternatively referred to as continuous time algorithms, are interpreted as dynamical closed loop control systems. The design is based on the control Liapunov function (CLF) method. For nonconvex scalar functions, the goal of these algorithms is to produce trajectories, starting from an arbitrarily chosen initial guess, that do not get stuck in local minima, thereby increasing the chances of converging to the global minimum.

Periodic solution to BAM-type Cohen-Grossberg neural network with time-varying delays

By using the continuation theorem of Mawhin’s coincidence degree theory and the Liapunov functional method, some sufficient conditions are obtained to ensure the existence, uniqueness and the global exponential stability of the periodic solution to the BAM-type Cohen-Grossberg neural networks involving timevarying delays.

UNIFORM DISSIPATIVENESS, ROBUST SYNCHRONIZATION AND LOCATION OF THE ATTRACTOR OF PARAMETRIZED NONAUTONOMOUS DISCRETE SYSTEMS

In this series of papers, we study issues related to the synchronization of two coupled chaotic discrete systems arising from secured communication. The first part deals with uniform dissipativeness with respect to parameter variation via the Liapunov direct method. We obtain uniform estimates of the global attractor for a general discrete nonautonomous system, that yields a uniform invariance principle in the autonomous case. The Liapunov function is allowed to have positive derivative along solutions of the system inside a bounded set, and this reduces substantially the difficulty of constructing a Liapunov function for a given system. In particular, we develop an approach that incorporates the classical Lagrange multiplier into the Liapunov function method to naturally extend those Liapunov functions from continuous dynamical system to their discretizations, so that the corresponding uniform dispativeness results are valid when the step size of the discretization is small. Applications to the discretized Lorenz system and the discretization of a time-periodic chaotic system are given to illustrate the general results. We also show how to obtain uniform estimation of attractors for parametrized linear stable systems with nonlinear perturbation.

Stabilization of feedback control and stabilizability optimal solution for nonlinear quadratic problems

This study refers to minimization of quadratic functionals in infinite time. The coefficients of the quadratic form are quadratic matrix, function of the state variable. Dynamic constraints are represented by bilinear differential systems of the form x ˙ = A ( x ) x + B ( x ) u , x ( 0 ) = x 0 . One selects an adequate factorization of A( x) such that the analyzed system should be controllable. Employing the Hamilton–Jacobi equation it results the matrix algebraic equation of Riccati associated to the optimum problem. The necessary extremum conditions determine the adjoint variables λ and the control variables u as functions of state variable, as well as the adjoint system corresponding to those functions. Thus one obtains a matrix differential equation where the solution representing the positive defined symmetric matrix P( x), verifies the Riccati algebraic equation. The stability analysis for the autonomous systems solution resulting for the determined feedback control is performed using the Liapunov function method. Finally we present certain significant cases.

Global Behaviors of a Chemostat Model with Delayed Nutrient Recycling and Periodically Pulsed Input

The dynamic behaviors in a chemostat model with delayed nutrient recycling and periodically pulsed input are studied. By introducing new analysis technique, the sufficient and necessary conditions on the permanence and extinction of the microorganisms are obtained. Furthermore, by using the Liapunov function method, the sufficient condition on the global attractivity of the model is established. Finally, an example is given to demonstrate the effectiveness of the results in this paper.

Global Stability Analysis of Multimedia Systems

A result of global stability for multimedia systems is given by means of the Liapunov functional method.

Periodic solution to BAM neural network with delays on time scales

In this paper, by using the theory of calculus on time scales, contraction mapping principle and Liapunov functional method, some sufficient conditions are obtained to ensure the existence and global exponential stability of periodic solution to BAM neural network with delays on time scales. This paper unifies the periodic discrete-time and continuous-time BAM neural networks under the same framework.

Global behaviors of Monod type chemostat model with nutrient recycling and impulsive input

In this paper, we consider the global behaviors of Monod type chemostat model with nutrient recycling and impulsive input. By introducing a new study method, the sufficient and necessary conditions on the permanence and extinction of the microorganisms are obtained. Furthermore, by using the Liapunov function method, the sufficient condition on the global attractivity of the system is established. Lastly, an example is given, the numerical simulation shows that if only the system is permanent, then it also is globally attractive.