Liapunov Function Method Research Articles

Dynamic behavior for a nonautonomous SIRS epidemic model with distributed delays

A nonautonomous SIRS epidemic model with distributed delay is investigated. Two new threshold values, R ∗ and R ∗ are derived. The model is permanent as R ∗ > 1 and R ∗ < 1 implies the extinction of the disease. Using the Liapunov functional method, global behavior of the model is studied.

Global exponential stability of delayed BAM network on time scale

Some sufficient conditions are derived to ensure the global exponential stability of delayed bi-directional associative memory (BAM) neural network on time scale, using the time scale calculus theory and the Liapunov functional method. The conditions possess highly important significance and can be easily checked in practice by simple algebraic methods. This is the first time applying the time scale calculus theory to unify and improve discrete-time and continuous-time bi-directional associate memory neural network under the same framework.

Stability of Stochastic Delay Differential Equation with a Small Parameter

Stochastic delay differential equations with wideband noise perturbations is considered. First it is shown that the perturbed system converges weakly to a stochastic delay differential equation driven by a Brownian motion. Stability and asymptotic properties of stochastic delay differential equations with a small parameter are developed. It is shown that the properties such as stability, recurrence, etc., of the limit system with time lag is preserved for the solution x ε(·) of the underlying delay equation for ε > 0 small enough. Perturbed Liapunov function method is used in the analysis.

Almost Sure Stability of a Moving Elastic Band

In this paper, the stochastic stability problem of a moving elastic band subjected to action in-plane acting forces is investigated. Each force consists of a constant part and a time-dependent zero mean stochastic function. By using the direct Liapunov functional method, almost sure asymptotic stability conditions are obtained as the function of stochastic process variance, damping coefficient, and geometric and physical parameters of the band. Numerical calculations are performed for infinite mode and compared with known results. Almost sure stability regions are shown for infinite and first mode the two-dimensional density probability function, and for higher modes when the edge load Gaussian or harmonic process is known.

Exponential decay estimates for time-delayed Benjamin–Bona–Mahony equations

In this article, we consider Benjamin–Bona–Mahony equation with a time delay. By using the Liapunov function method, we show that the time-delayed Benjamin–Bona–Mahony equation is exponentially decay if the delay parameter is sufficiently small.

Integral stability criteria of nonlinear differential systems

The notions of equi-integral and ϕ 0 -stability for systems of ordinary differential equations (ODEs) were introduced and extended. The notions of equi-integral stability are extended to the so-called equi-integral ϕ 0 -stability of perturbed system of (ODEs). Our technique depends on cone-valued Liapunov function method.

Primary and secondary frequency regulation in a longitudinal hydrothermal system by Liapunov method

The problem of primary and secondary frequency regulation of a multi-machine power system is analysed with a Liapunov function method based on a quadratic performance index. The main purpose is to determine the best set of parameters of primary and secondary regulation according to an optimisation criterion. The objective consists of achieving a settling time as short as possible which is relevant in a longitudinal system, whose structure includes long transmission lines with a reduced number of loops. Frequency and turbine valve positions are used as state variables, and optimal control parameters for primary and secondary regulation are used as control variables. Performance index behaviour under different frequency control schemes is then used to determine the optimal values to set the droops and gains of primary and secondary regulation. Tests for different disturbance locations and selection of regulating units are simulated on a model of the Chilean Central Interconnected System (CIS). It is shown that significant improvements can be achieved with the proposed method.

Phase transitions in a coupled viscoelastic system with periodic initial-boundary condition: (I) Existence and uniform boundedness

This paper focuses on the phase transitions of a 2$\times$2 system of mixed type for viscosity-capillarity with periodic initial-boundary condition in a viscoelastic material. By the Liapunov functional method, we prove the existence, uniqueness, regularity and uniform boundedness of the solution. The results are correct even for large initial data.

Control of multi-node mobile communications networks with time-varying channels via stability methods

Consider a communications network consisting of mobiles and random external data processes, each destined to a particular destination. Each mobile can serve as a node in the multi-hop path from source to destination. At each mobile the data is queued according to the source destination pair. The quality of the connecting channels are randomly varying. Time is divided into small scheduling intervals. At the beginning of each interval, the channels are estimated and this information is used for the decisions concerning allocation of transmission power and/or time, bandwidth, and perhaps antennas, in a queue and channel-state dependent way. Under a natural (and "almost" necessary) "average flow" condition, stochastic stability methods are used to develop scheduling policies that assure stability. The policies are readily implementable and allow a range of tradeoffs between current rates and queue lengths, under very weak conditions. Because of the non-Markovian nature of the problem, we use the perturbed Stochastic Liapunov function method. The choice of Liapunov function allows a choice of the effective performance criteria. All essential factors are incorporated into a "mean rate" function, so that the results cover many different systems. Extensions concerning acknowledgments, multicasting, non-unique routes, and others are given to illustrate the versatility of the method, and a useful method for getting the a priori routes is discussed.

Spatio-temporal chaotic synchronization for modes coupled two Ginzburg-Landau equations

On the basis of numerical computation, the conditions of the modes coupling are proposed, and the high-frequency modes are coupled, but the low frequency modes are uncoupled. It is proved that there exist an absorbing set and a global finite dimensional attractor which is compact and connected in the function space for the high-frequency modes coupled two Ginzburg-Landau equations (MGLE). The trajectory of driver equation may be spatio-temporal chaotic. One associates with MGLE, a truncated form of the equations. The prepared equations persist in long time dynamical behavior of MGLE. MGLE possess the squeezing properties under some conditions. It is proved that the complete spatio-temporal chaotic synchronization for MGLE can occur. Synchronization phenomenon of infinite dimensional dynamical system (IFDDS) is illustrated on the mathematical theory qualitatively. The method is different from Liapunov function methods and approximate linear methods.

P-Moment stability of stochastic differential equations with impulsive jump and Markovian switching

This paper introduces some new concepts of p-moment stability for stochastic differential equations with impulsive jump and Markovian switching. Some stability criteria of p-moment stability for stochastic differential equations with impulsive jump and Markovian switching are obtained by using Liapunov function method. An example is also discussed to illustrate the efficiency of the obtained results.

Partial synchronization between different systems

A new method for partial synchronization between different systems was obtained. The definition of partial synchronization under which the problem works is given. The stability of the method is analyzed by the Liapunov function method and the condition of choosing the control term is derived. The reliability of this method is proved by some numerical examples, in which the dynamical behaviors of the synchronized systems are observed and it is found that whatever state the response system is partial synchronization can be always achieved by adding some proper control term.

LMS Algorithms for Tracking Slow Markov Chains With Applications to Hidden Markov Estimation and Adaptive Multiuser Detection

This paper analyzes the tracking properties of the least mean squares (LMS) algorithm when the underlying parameter evolves according to a finite-state Markov chain with infrequent jumps. First, using perturbed Liapunov function methods, mean-square error estimates are obtained for the tracking error. Then using recent results on two-time-scale Markov chains, mean ordinary differential equation and diffusion approximation results are obtained. It is shown that a sequence of the centered tracking errors converges to an ordinary differential equation. Moreover, a suitably scaled sequence of the tracking errors converges weakly to a diffusion process. It is also shown that iterate averaging of the tracking algorithm results in optimal asymptotic convergence rate in an appropriate sense. Two application examples, analysis of the performance of an adaptive multiuser detection algorithm in a direct-sequence code-division multiple-access (DS/CDMA) system, and tracking analysis of the state of a hidden Markov model (HMM) with infrequent jumps, are presented.

A diffusive stage-structured model in a polluted environment

In this paper we discuss a spatial nonhomogeneous stage-structured self-diffusion model in the polluted environment. The global stability of the equilibrium can be derived based on the stability of the corresponding ordinary differential system using Liapunov function method. The effect of diffusion on the stability of the system is also investigated.

Global asymptotic stability conditions of delayed neural networks

Utilizing the Liapunov functional method and combining the inequality of matrices technique to analyze the existence of a unique equilibrium point and the global asymptotic stability for delayed cellular neural networks (DCNNs), a new sufficient criterion ensuring the global stability of DCNNs is obtained. Our criteria provide some parameters to appropriately compensate for the tradeoff between the matrix definite condition on feedback matrix and delayed feedback matrix. The criteria can easily be used to design and verify globally stable networks. Furthermore, the condition presented here is independent of the delay parameter and is less restrictive than that given in the references.

On practical stability of perturbed differential systems

The notions of practical stability of systems of ordinary differential equations (ODE) was introduced. In this paper, we will extend practical stability notion to a new type of stability called practical ϕ 0-stability of perturbed systems of ordinary differential equations, and given Some criteria and results. Our technique depends on cone-vector Liapunov function method.

On cone perturbing Liapunov function for impulsive differential systems

The concept of ϕ 0-stability and ϕ 0-boundedness recently were introduced for systems of ordinary differential equations (ODEs). Perturbing Liapunov function method was discussed for systems of ODEs and extend to systems of functional differential equations (FDEs). In this paper, we extend these notions to impulsive systems of differential equations via cone perturbing Liapunov function method.

Stability Analysis of Nonlinear Systems via Descriptor Equations

The purpose of this paper is to emphasize that for nonlinear systems, analysis and synthesis methods based on descriptor equations must be developed. First, two illustrative examples show that descriptor equations are superior to state-space equations in representing nonlinear systems as models for analysis and synthesis. Then, a new stability condition for nonlinear descriptor systems is derived on the basis of Liapunov function method. As an application of the method, an LMI condition is obtained for the stability of Lur'e-type feedback systems whose linear parts include direct paths.

Global behaviors of a generalized periodic impulsive Logistic system with nonlinear density dependence

The global behaviors of a generalized periodic impulsive Logistic system with nonlinear density dependence are studied. Conditions for the existence and global attractivity of positive periodic solution are obtained via the method of comparison and Liapunov function. The corresponding results for the periodic impulsive Logistic system, which are dependent on solving the system, are extended.

Permanence of discrete-time Kolmogorov systems for two species and saturated fixed points

This paper considers the dynamics of a discrete-time Kolmogorov system for two-species populations. In particular, permanence of the system is considered. Permanence is one of the concepts to describe the species' coexistence. By using the method of an average Liapunov function, we have found a simple sufficient condition for permanence of the system. That is, nonexistence of saturated boundary fixed points is enough for permanence of the system under some appropriate convexity or concavity properties for the population growth rate functions. Numerical investigations show that for the system with population growth rate functions without such properties, the nonexistence of saturated boundary fixed points is not sufficient for permanence, actually a boundary periodic orbit or a chaotic orbit can be attractive despite the existence of a stable coexistence fixed point. This result implies, in particular, that existence of a stable coexistence fixed point is not sufficient for permanence.