Equiasymptotic Stability Research Articles

Global asymptotic stability and equiasymptotic stability for a time-varying phytoplankton–zooplankton–fish system

In this paper, we consider a three-species system which consists of phytoplankton, zooplankton and fish. It is well known that zooplankton are harvested by other creatures besides fish, and it has been reported that zooplankton can absorb energy from other micro-organisms besides phytoplankton in recent decades. For this reason, we add the parts of extra energy source and harvesting effect in the model. Taking account of seasonal variation which can bring affects to many aspects of plankton population, we assume that some factors about the phytoplankton and the zooplankton are time-dependent. In particular, since the environment does not change regularly, we does not require these time-varied factors are periodic, which is quite different with most ecological models. The purpose of this paper is to present sufficient conditions which guarantee that the equilibrium of this three-species system is globally asymptotically stable. Moreover, we will show that the equilibrium is equiasymptotically stable under relatively weaker conditions.

Exponential stability for set dynamic equations on time scales

A new theory known as set dynamic equations on time scales has been built. The criteria for the equistability, equiasymptotic stability, uniform and uniformly asymptotic stability were developed in Hong (2010) [1]. In this paper, we consider the exponential stability, exponentially asymptotic stability, uniform and uniformly exponentially asymptotic stability for the trivial solution of set dynamic equations on time scales by using Lyapunov-like functions.

Stability criteria for set dynamic equations on time scales

Very recently, a new theory known as set dynamic equations on time scales has been built. In this paper, notions of stability for the solutions of set dynamic equations on time scales, using Lyapunov-like functions are considered. Criteria for the equistability, equiasymptotic stability, uniform and uniform asymptotic stability are developed.

On equiasymptotic stability of solutions of doubly-periodic impulsive systems

We consider a system of ordinary differential equations with pulse action at fixed times that admits the trivial solution. We establish sufficient conditions for the equiasymptotic stability of the trivial solution.

Stability criteria for set differential equations

Notions of stability for the solutions of set differential equations, using Lyapunov-like functions are considered. Criteria for the equistability, equiasymptotic stability, uniform and uniform asymptotic stability are presented.

Partial equiasymptotic stability of nonlinear dynamic systems

For nonlinear nonautonomous dynamic systems, partial equiasymptotic stability of the zero equilibrium position, equiasymptotic stability of the "partial" equilibrium position, and partial equiasymptotic stability of the "partial" equilibrium position are studied by the Lyapunov functions method. Examples are given.

Partial equiasymptotic stability in differential equations with delay

A system of functional differential equations with delay is considered. It is supposed that this system has a zero solution. Definitions of its partial stability, partial asymptotical stability, and partial equiasymptotical stability are given. Theorems on the partial equiasymptotical stability are formulated and proved.

On the Partial Equiasymptotic Stability in Functional Differential Equations

A system of functional differential equations with delay dz/dt=Z(t,zt), where Z is the vector-valued functional is considered. It is supposed that this system has a zero solution z=0. Definitions of its partial stability, partial asymptotical stability, and partial equiasymptotical stability are given. Theorems on the partial equiasymptotical stability are formulated and proved.

Equiasymptotic stability in delay differential-functional systems

According to [1, 2], a differential equation with deviating argument is an equation that contains the values of the unknown function and its derivatives at different values of the argument t (time). Such an equation is referred to as a delay equation if the value of the higher-order derivative at each time t is expressed via the values of lower-order derivatives at earlier instants t + 0, where 0 0, by C we denote the space of continuous functions on the closed interval [ -h ; 0] with values in R ~. For ~ E C, we set II~lI sUP-h 0, and t is a given number, t E [0; A), then, following [4], by xt we denote the restriction of the function x to the closed interval [t h; t]. Therefore, xt is the element of C given by the formula xt(O) = x(t + O) for h < 0 < 0. Let us consider the system of differential-functional equations

On equi-asymptotic stability with respect to part of the variables

A system of equations of perturbed motion in which the right-hand sides are almost periodic functions of time is considered. A sufficient condition for the trivial solution of the system to be equi-asymptotically stable with respect to part of the variables is proved.

Asymptotic stability of integral sets

We prove theorems on asymptotic, equiasymptotic, and uniform asymptotic stability of the integral sel of a nonautonomous system of ordinary differential equations.

Equiasymptotic stability of integral sets

Equiasymptotic stability of integral sets

Asymptotic stability for a homogeneous singularly perturbed system of differential equations with unbounded delay

Sufficient conditions for equiasymptotic stability of the null solution of a homogeneous singularly perturbed system of differential equations with unbounded delay are obtained.

Asymptotic stability for a homogeneous singularly perturbed system of differential equations with unbounded delay

Sufficient conditions for equiasymptotic stability of the null solution of a homogeneous singularly perturbed system of differential equations with unbounded delay are obtained.

Perturbing Lyapunov functions and global results

THE PROOFS of many results in stability theory depend on dividing the vicinity of some kind of invariant set (or other convenient set) into suitable subsets and then showing that the solutions cannot leave such sets. This allows us to obtain global results in terms of arbitrary sets [2] and Lyapunov functions that satisfy some weaker assumptions. It is also known [3,4] that by perturbing Lyapunov functions and employing a family of Lyapunov functions, nonuniform stability results can be proved under weaker hypothesis. The main idea of perturbing the Lyapunov function is to make use of a Lyapunov function that may not be satisfying all the desired conditions and to perturb it in such a way, one gets a Lyapunov function that is useful in the investigation of stability of different systems. In this paper, we shall present two global theorems of a general character using Lyapunov functions and certain arbitrary sets. These results are then applied to obtain equi-stability and equi-asymptotic stability, showing the nonuniform behavior proved with less requirements. Our approach enlarges the class of useful Lyapunov functions and offers more flexibility. These global results can be utilized to prove several known results in stability and boundedness.

Asymptotic stability criteria for delay-differential equations

SynopsisThis paper is concerned with the problem of showing uniform stability and equiasymptotic stability of thezero solution of functional differential equations with either finite or infinite delay. The investigations are based on Liapunov's direct method and attention is focused on those equations whose right-hand sides are unbounded for bounded state variables.

On partial stability of differential equations with time delay

We investigate stability, uniform stability and equi-asymptotic stability with respect to the x-components and y-components of a differential equation with time delay. We also obtain necessary and sufficient conditions for the generalized asymptotic stability of the exponential type with respect to the components which generalizes the work of Corduneanu[3]. We make use of Lyapunov functionals and differential inequalities in our study.

Stability and existence of a periodic solution

Many authors have discussed the existence of a periodic solution of a periodic system of differential equations under the assumption that the system has a bounded solution which has some kind of stability. LaSalle [Z] has discussed a more general case by considering an asymptotic stability property of motions. Hale [2] and the author [3] have discussed the existence of a periodic solution of functional-differential equations (more generally, the existence of an almost periodic solution) under the assumption that the system has a bounded solution which is uniformly asymptotically stable in the large by applying Liapunov’s second method. Jones [4] also has discussed the existence of a periodic solution by assuming that a set has an asymptotic stability property in some sense. Recently, Sell [5] has shown the existence of a periodic solution of a periodic system of period w under the assumption that the system has a bounded solution which is uniformly asymptotically stable or uniformly asymptotically stable in the large. However, Sell’s definition of uniform asymptotic stability differs from the usual one. (See [a). His type of stability is weaker than the uniform asymptotic stability in the usual sense and it is stronger than the asymptotic stability. Actually, as is seen below, it is also stronger than the equiasymptotic stability. Sell shows that there exists a periodic solution of period KW for some integer K > 1. In this paper, we shall show that Sell’s concept is stronger than the equiasymptotic stability and that for a periodic system Sell’s concept for uniform asymptotic stability of a bounded solution is equivalent to the usual concept, and consequently we have a periodic solution of period w under the assumption that a bounded solution is uniformly asymptotically

Stability theory and Lyapunov's second method

1. Introduct ion In two papers appearing in t949 and t956, MASSERA [8, 9] made a number of significant advances in LYAPUNOV'S second method and the theory of stability of ordinary differential equations. He extended the work of LYAPUNOV [31, MALKIN [4--71 and others and arrived at both necessary and sufficient conditions for stability in terms of a class of Lyapunov functions. However, even though necessary and sufficient conditions are known for various types of stability, it is still a serious problem in applications of this method to choose the Lyapunov function which would imply stability of a given type, say type X. From the applied point of view, this problem is far from solved; however, we are able to move closer to a solution in two ways in this paper. The first is to define properties of the Lyapunov functions (along the solutions of the differential equation) that are equivalent to the various types of stability considered in this paper. Thus a correspondence between stability of type X ' a n d property 2 of the Lyapunov functions is established. Then it is shown that the null solution of (t) is stable of type X if and only if every Lyapunov function which lies in 93~c has property 2 (Theorem t). In this way the problem of choosing a Lyapunov function V to test for stability of type X is eliminated; however, it now is necessary to determine whether or not V has property 2. The second way is to seek sufficient conditions on the generalized total derivative V' of the Lyapunov function V (with respect to (t)) which insure that V has property 2. ~[n particular, we are able to generalize the results of LYAPUNOV, MALKIN and MASSERA and get weaker conditions for Lyapunov stability, uniform stability, equi-asymptotic stability and uniform-asymptotic stability in Sections 4 and 5. The followingol~hma, which can easily be verified, will prove to be helpful.

On the equiasymptotic stability in the large

On the equiasymptotic stability in the large