差分方程解的稳定性、有界性及概周期解的存在性

作者通过Liapunov泛函建立了一类高维差分方程解一致稳定、一致渐近稳定及指数渐近稳定的充要条件. 此外, 作者还证明了解的一致渐近稳定性蕴含解的有界性, 同时也给出了概周期差分方程存在概周期解的一个充分条件.     

Stability theory for a two-dimensional channel

A scheme for deriving conditions for the nonlinear stability of an ideal or viscous incompressible steady flow in a two-dimensional channel that is periodic in one direction is described. A lower bound for the main factor ensuring the stability of the Reynolds–Kolmogorov sinusoidal flow with no-slip conditions (short wavelength stability) is improved. A condition for the stability of a vortex strip modeling Richtmyer–Meshkov fluid vortices (long wavelength stability) is presented.     

Mittag-Leffler-Ulam stabilities of fractional evolution equations

In this paper, we present and discuss four types of Mittag-Leffler-Ulam stability: Mittag-Leffler-Ulam-Hyers stability, generalized Mittag-Leffler-Ulam-Hyers stability, Mittag-Leffler-Ulam-Hyers-Rassias stability and generalized Mittag-Leffler-Ulam-Hyers-Rassias stability for a fractional evolution equation in Banach spaces.     

STABILITY OF THEORETICAL SOLUTION AND NUMERICAL SOLUTION OF NONLINEAR DIFFERENTIAL EQUATIONS WITH PIECEWISE DELAYS

This paper is concerned with the stability of theoretical solution and numerical solutionof a class of nonlinear differential equations with piecewise delays.At first,a sufficientcondition for the stability of theoretical solution of these problems is given,then numericalstability and asymptotical stability are discussed for a class of multistep methods whenapplied to these problems.     

Stability of Liénard Equation

We consider a balance stability problem for the second order nonlinear differential equations of the Liénard type. Investigations are carried out by means of constant-sign Lyapunov functions for problems of stability, asymptotical stability (local and global), and instability. We implicitly formulate a method of construction of constant-sign functions suitable for solving problems of motion stability. Special attention is paid to a problem of non-asymptotical stability, where we demonstrate possibilities of new assertions that rely upon a usage of constant-sign Lyapunov functions.     

Generalizations of a theorem of Rolewicz

The aim of this article is to give a unified treatment for the theorems of Rolewicz and Neerven type for uniform exponential stability of evolution families. We obtain necessary and sufficient conditions for uniform exponential stability of evolution families, generalizing a stability theorem due to Rolewicz and we present a new proof for the Rolewicz theorem, based on the theory of Banach function spaces. Finally, we apply our results and we deduce a generalization for a classical stability theorem due to Przyluski and Rolewicz.     

Stability of functional equations in single variable

This paper discusses Hyers-Ulam stability for functional equations in single variable, including the forms of linear functional equation, nonlinear functional equation and iterative equation. Surveying many known and related results, we clarify the relations between Hyers-Ulam stability and other senses of stability such as iterative stability, continuous dependence and robust stability, which are used for functional equations. Applying results of nonlinear functional equations we give the Hyers-Ulam stability of Böttcher's equation. We also prove a general result of Hyers-Ulam stability for iterative equations.     

Analytical and numerical methods for the stability analysis of linear fractional delay differential equations

In this paper, several analytical and numerical approaches are presented for the stability analysis of linear fractional-order delay differential equations. The main focus of interest is asymptotic stability, but bounded-input bounded-output (BIBO) stability is also discussed. The applicability of the Laplace transform method for stability analysis is first investigated, jointly with the corresponding characteristic equation, which is broadly used in BIBO stability analysis. Moreover, it is shown that a different characteristic equation, involving the one-parameter Mittag-Leffler function, may be obtained using the well-known method of steps, which provides a necessary condition for asymptotic stability. Stability criteria based on the Argument Principle are also obtained. The stability regions obtained using the two methods are evaluated numerically and comparison results are presented. Several key problems are highlighted.     

无限维关联系统的弦稳定性

对一类无限维关联系统引入弦稳定概念。系统弦稳定意谓着,当关联系统的初始状态为有界时,对任意时刻系统的状态也是有界的。本文将向量V函数法推广到无限维系统中,得到了关联系统渐近弦稳定的充分条件,克服了以前的方法在处理非线性系统的稳定性问题上的困难,扩大了系统稳定的参数范围。     

Dynamics of the density dependent predator-prey system with Beddington-DeAngelis functional response

Two models of a density dependent predator-prey system with Beddington-DeAngelis functional response are systematically considered. One includes the time delay in the functional response and the other does not. The explorations involve the permanence, local asymptotic stability and global asymptotic stability of the positive equilibrium for the models by using stability theory of differential equations and Lyapunov functions. For the permanence, the density dependence for predators is shown to give some negative effect for the two models. Further the permanence implies the local asymptotic stability for a positive equilibrium point of the model without delay. Also the global asymptotic stability condition, which can be easily checked for the model is obtained. For the model with time delay, local and global asymptotic stability conditions are obtained.     

On the stability of inhomogeneously viscoelastic reinforced bars

There is investigated the stability of inhomogeneously ageing reinforced viscoelastic bars. It is assumed that the strains and stresses in the reinforcement are related by Hooke's law. The properties of the matrix material are described by equations of the theory of viscoelasticity of inhomogeneously ageing solids /1,2/. Under different boundary conditions for the ends of the bar and loading methods an expression is set up for the critical force in stability problems in an infinite time interval. The stability definition taken corresponds to the Liapunov stability definition for the motion of dynamical systems. Estimates of the critical time when the magnitude of the deflection of a viscoelastic bar reaches a given value are obtained for stability problems in a finite time interval. The formulation for the stability problem in a finite time interval starts from the definition of stability of motion of dynamical systems by taking its beginning from the Chetaev work. The dependence of the critical time on the inhomogeneity and the reinforcing parameter is investigated numerically. The stability of viscoelastic unreinforced bars was studied in /3,4/, A survey and bibliography of research associated with the stability problem for viscoelastic bars are available in /5–8/.     

What price stability? Social welfare in matching markets

In two-sided matching markets, stability can be costly. We define social welfare functions for matching markets and use them to formulate a definition of the price of stability. We then show that it is common to find a price tag attached to stability, and that the price of stability can be substantial. Therefore, when choosing a matching mechanism, a social planner would be well advised to weigh the price of stability against the value of stability, which varies from market to market.     

Stability analysis and stabilization of LPV systems with jumps and (piecewise) differentiable parameters using continuous and sampled-data controllers

Linear Parameter-Varying (LPV) systems with jumps and piecewise differentiable parameters is a class of hybrid LPV systems for which no tailored stability analysis and stabilization conditions have been obtained so far.¹ We fill this gap here by proposing an approach based on a clock- and parameter-dependent Lyapunov function yielding stability conditions under both constant and minimum dwell-times. Interesting adaptations of the latter result consist of a minimum dwell-time stability condition for uncertain LPV systems and LPV switched impulsive systems. The minimum dwell-time stability condition is notably shown to naturally generalize and unify the well-known quadratic and robust stability criteria all together. Those conditions are then adapted to address the stabilization problem via timer-dependent and a timer- and/or parameter-independent (i.e. robust) state-feedback controllers, the latter being obtained from a relaxed minimum dwell-time stability condition involving slack-variables. Finally, the last part addresses the stability of LPV systems with jumps under a range dwell-time condition which is then used to provide stabilization conditions for LPV systems using a sampled-data state-feedback gain-scheduled controller. The obtained stability and stabilization conditions are all formulated as infinite-dimensional semidefinite programming problems which are then solved using sum of squares programming. Examples are given for illustration.     

New stability criteria for a class of neutral systems with discrete and distributed time-delays: an LMI approach

In this paper, a problem of the asymptotic stability for a class of neutral systems with multiple discrete and distributed time-delays is considered. Lyapunov stability theory is applied to guarantee the stability for the systems. New discrete-delay-independent and discrete-delay-dependent stability conditions are derived in terms of the spectral radius and linear matrix inequality. By mathematical analysis, the stability criteria are proved to be less conservative than the ones reported in the current literatures. A numerical example is given to illustrate the availability of the proposed results.     

带跳随机微分方程的Euler-Maruyama方法的几乎处处指数稳定性和矩稳定性

本文首先研究了一维带跳随机微分方程的指数稳定性,并证明Euler-Maruyama(EM)方法保持了解析解的稳定性.其次,研究了多维带跳随机微分方程的稳定性,证明若系数满足全局Lipchitz条件,则EM方法能够很好地保持解析解的几乎处处指数稳定性、均方指数稳定性.最后,给出算例来支持所得结论的正确性.     

Stability analysis of diffusive predator-prey model with modified Leslie-Gower and Holling-type III schemes

The stability of a diffusive predator-prey model with modified Leslie-Gower and Holling-type III schemes is investigated. A threshold property of the local stability is obtained for a boundary steady state, and sufficient conditions of local stability and un-stability for the positive steady state are also obtained. Furthermore, the global asymptotic stability of these two steady states are discussed. Our results reveal the dynamics of this model system.     

Aggregation and stability analysis of nonlinear complex systems

The problem of stability of large-scale systems in critical cases is investigated. New form of aggregation for essentially nonlinear complex systems is suggested. With the help of this form the sufficient conditions of asymptotic stability are determined. The results obtained are used for the stability analysis of complex systems by the nonlinear approximation and for the investigation of absolute stability conditions for a certain class of nonlinear systems.     

Strict stability in terms of two measures for impulsive differential equations with ‘supremum’

Strict stability for a nonlinear system of impulsive differential equations with ‘supremum’ is defined and studied. Razhumikhin method with piecewise continuous scalar Lyapunov functions and comparison results for scalar impulsive differential equations are the bases of the main proofs. To unify a variety of stability concepts and to offer a general framework for the investigation of the stability theory, the notion of stability in terms of two measures has been applied. An example illustrating the usefulness of the obtained sufficient conditions is also included.     

NONLINEAR STABILITY OF NATURAL RUNGE-KUTTA METHODS FOR NEUTRAL DELAY DIFFERENTIAL EQUATIONS

This paper first presents the stability analysis of theoretical solutions for a class of nonlinear neutral delay-differential equations (NDDEs). Then the numerical analogous results, of the natural Runge-Kutta (NRK) methods for the same class of nonlinear NDDEs, are given. In particular, it is shown that the (k, l)-algebraic stability of a RK method for ODEs implies the generalized asymptotic stability and the global stability of the induced NRK method.