Stability in first approximation of stochastic systems with after effect : PMM vol. 40, n 6, 1976, pp. 1116–1121

The theorem on existence of the Liapunov functionals and the theorem on stability in first approximation for a stochastic differential equation with aftereffect are proved.The suggestion of the replacement of Liapunov functions by functionals [1] in the investigation of the stability of ordinary differential equations with lag, has been widely utilized in dealing with determinate systems, as well as in the case of linear and nonlinear stochastic systems (see e. g. [2 – 11]). Results concerning the stability in the first approximation were obtained for stochastic systems in [12 – 18] and others. Use of Liapunov functionals for the differential equations with aftereffect was first encountered in [1, 19, 20] where the inversion theorems were proved and conditions for the stability in first approximation were obtained.Below a stochastic differential equation with aftereffect is investigated where the random perturbations represent an arbitrary process with independent increments.     

On global existence and attractivity results for nonlinear functional integral equations

In this paper two existence results concerning the global attractivity and global asymptotic attractivity for a certain functional nonlinear integral equation are proved. Our existence results include several existence as well as attractivity results obtained earlier by Banas and Dhage (2008) [1], Hu and Yan (2006) [3], Dhage (2009) [15] and Banas and Rzepka (2003) [7] as special cases under some weaker Lipschitz conditions. A measure theoretic fixed point theorem of Dhage (2008) [6] is used in formulating our main results and the characterizations of solutions are obtained in the space of functions defined, continuous and bounded on unbounded intervals.     

Two families of liapunov functions for functional differential systems

Two families of Liapunov functions are employed to study the global stability and boundedness of functional differential systems. New stability and boundedness theorems are obtained. Applications of these theorems to some nonlinear differential systems with infinite delay are discussed. Project supported by the National Science Foundation of China Under Grants 69871005     

Nuovi criteri per l'esistenza globale in futuro dei moti dei sistemi olonomi scleronomi

Summary By means of the comparison method and an appropriate choice of the Liapunov function, new criteria for the global existence in the future of the motions of several classes of holonomic scleronomic systems are obtained. The main advantage of these criteria respect to those provided in [1], [2]is that the potential energy of the mechanical system need not be bounded from below, as required in the above-mentioned papers, where the total energy of the system is chosen as Liapunov function.     

关于非线性系统的非常稳定性及其应用

本文将C.V.Pao研究Liapunov稳定性的内积法改进、推广和发展,研究了非线性系统的非常稳定性、平衡位置的存在唯一性和Liapunov稳定性.将主要结果直接应用到非线性周期系统的稳态振荡的判定.     

On the stability of a satellite cylindrical precession in an elliptic orbit : PMM vol. 40, n 6, 1976, pp. 1040–1047

The stability of motion of a dynamically symmetric satellite with respect to its center of mass in a central Newtonian gravitational field is investigated. The satellite is a solid body whose center of mass moves on an elliptic orbit. The particular case in which the satellite axis of symmetry is normal to the orbit plane (the so-called cylindrical precession [1, 2]) and its absolute angular velocity projection on the axis of symmetry is zero, is examined. Analytical and numerical methods are used. Regions of Liapunov instability and of stability in the first approximation are. obtained in the parameter space of the problem (the inertial parameter and the orbit eccentricity). Detailed nonlinear analysis is carried out in the latter, and the formal stability of the satellite cylindrical precession is proved. The question of stability for the majority of intial conditions is also considered [4].     

两类非线性系统的全局稳定性

本文利用 Liapunov 函数方法和论证系统正半轨线有界的 Shimanov区域方法,给出了两类非线性系统的零解为全局稳定的充分条件,并讨论了一些低阶实例,得到了较好的结果.     

两自由度非线性振动系统周期运动及其稳定性研究

运用Liapunov函数方法,对一类两自由度非线性振动系统周期运动及其稳定性进行了研究,得到了存在唯一渐近稳定的周期解的充分条件.     

Well-posedness and existence of bound states for a coupled Schrödinger-gKdV system

We derive some new results concerning the Cauchy problem and the existence of bound states for a class of coupled nonlinear Schrödinger-gKdV systems. In particular, we obtain the existence of strong global solutions for initial data in the energy space H¹(R)×H¹(R), generalizing previous results obtained in Tsutsumi (1993) [11], Corcho and Linares (2007) [13] and Dias et al. (submitted for publication) [14] for the nonlinear Schrödinger-KdV system.     

Existence,uniqueness, and stability of nonlinear evolution equations

In this paper we study existence, uniqueness, and stability of nonlinear evolution equations. We develop a new type of perturbation result for a C₀ semigroup in Banach space, where the nonlinear operators are not necessarily m-accretive or everywhere defined. Assuming that the semigroup has a smoothing property we obtain local existence, uniqueness and regularity results. We then establish a Liapunov theory which enables us to examine stability. To illustrate our theory several simple examples are presented.     

一类非线性系统周期解的存在唯一性

运用Liapunov函数方法,研究了一类四阶非线性系统,得到了该非线性系统存在唯一渐近稳定的周期解的充分条件.     

无限时滞泛函微分方程概周期解的存在唯一性

对于无限时滞泛函微分方程,利用Liapunov泛函的方法,研究了方程概周期解的存在性、唯一性问题,得到了便于应用的概周期解的存在性、唯一性判据.     

On the long-time dynamics of nonautonomous predator–prey models with mutual interference

The longtime behaviour of a nonautonomous bidimensional Hassell predator?Cprey model with mutual interference is investigated. The existence of an absorbing set in the phase space is shown, and necessary and sufficient conditions guaranteeing the nonlinear, global, asymptotic stability of the positive solutions have been found by using the Liapunov direct method.     

Asymptotic stability and smooth equivalence op ordinary equations : PMM vol.40, n 6, 1976, pp. 1048–1057

Under certain specified conditions the asymptotic stability is a coarse property [1],(i.e. addition of fairly smooth functions to the right-hand sides of equations, does not disturb the asymptotic stability). It is shown below that in this cage the unperturbed system is coarse in a more general sense, namely, any smooth system acted upon by fairly small smooth perturbations, can be returned to its unperturbed state by a smooth reversible transformation. The value and order of the perturbations and the domain of existence of the transformation are all estimated explicitly. The condition required for the above assertion to hold, is that of the existence of a Liapunov function admitting, together with its derivative, specified estimates. This requirement holds, in particular, in the case when the right-hand sides of the unperturbed system are homogeneous functions, the position of equilibrium is asymptotically stable, and its neighborhood contains no solutions bounded when −∞ <t < ∞ (see [1]). If the system is analytic, the requirement will hold in at least all critical cases investigated in which the asymptotic stability with t → ∞ or t → −∞ is fixed, since in these cases the Liapunov function will be analytic, or simply polynomial. It follows therefore from the theorem which we prove, that in all the cases in question, the system is reduced by a smooth transformation, to the polynomial form. If the unperturbed system is linear, then from the theorem proved follows a theorem on linearization appearing in [2]; if the system is nonlinear but of second order, a theorem from [3] ensues. The results obtained in this paper for the nonlinear autonomous systems are extended to the case when the perturbations are continuous and bounded functions of time. This makes possible the investigation of the dynamics of the process in the neighborhood of asymptotically stable equilibria and of periodic modes, ignoring a wide range of external perturbations.     

Local asymptotic stability for dissipative wave systems

We study the question of asymptotic stability, as time tends to infinity, of solutions of dissipative wave systems, governed by time-dependent nonlinear damping forces and by strongly nonlinear potential energies. This problem had been considered earlier for potential energies which arise from restoring forces, whereas here we allow as well for the effect of amplifying forces. Global asymptotic stability can then no longer be expected, and should be replaced by local stability. The conclusions are related to and supplement earlier work of Payne and Sattinger [7], who treated the nondissipative case, and of Hale [1], who showed the existence of connected global attractors.     

On liapunov functional in theory of stability of systems with time-lag

It is well known, that in the theory of stability in differential equations, Liapunov's second method may be the most important. The center problem of Liapunov's second method is construction of Liapunov function for concrete problems. Beyond any doubt, construction of Liapunov functions is an art. In the case of functional differential equations, there were also many attempts to establish various kinds of Liapunov type theorems. Recently Burton [2] presented an excellent theorem using the Liapunov functional to solve the asymptotic stability of functional differential equation with bounded delay. However, the construction of such a Liapunov functional is still very hard for concrete problems. In this paper, by utilizing this theorem due to Burton, we construct concrete Liapunov functional for certain and nonlinear delay differential equations and derive new sufficient conditions for asymptotic stability. Those criteria improve the result of literature [1] and they are with simple forms, easily checked and applicable.This project is supported by the National Natural Science Foundation of China.     

Exponential stability of periodic neural networks with impulsive effects and time-varying delays

By employing Young inequality and constructing suitable Liapunov functions, we investigate the existence and globally exponential stability of periodic neural networks with impulses and time-varying delays. The results extend and improve some earlier ones [1], [5] and [12]. An illustrative example and simulations are given to show the validity of the main results.     

Sull'esistenza globale e sulla limitatezza dei moti dei Sistemi olonomi scleronomi

Summary Sufficient conditions for the global existence in the future and the partial boundedness of the motions of holonomic scleronomic mechanical systems, using the comparison method introduced by R. Conti [1] and talcing as «Liapunov function» the total energy, are obtained, generalizing well known results [2, 3, 4].     

Global exponential stability of a class of retarded impulsive differential equations with applications

This paper studies the dynamics of a class of retarded impulsive differential equations (IDE), which generalizes the delayed cellular neural networks (DCNN), delayed bidirectional associative memory (BAM) neural networks and some population growth models. Some sufficient criteria are obtained for the existence and global exponential stability of a unique equilibrium. When the impulsive jumps are absent, our results reduce to its corresponding results for the non-impulsive systems. The approaches are based on Banach’s fixed point theorem, matrix theory and its spectral theory. Due to this method, our results generalize and improve many previous known results such as [3], [5], [6], [9], [17], [18], [23], [32], [38], [43], [51], [52]. Some examples are also included to illustrate the feasibility and effectiveness of the results obtained.     

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.