About stability of difference equations with continuous time and fading stochastic perturbations

The long term behavior of solutions of difference equations with continuous time and fading stochastic perturbations is investigated. It is shown that if the level of stochastic perturbations fades on the infinity, for instance, if it is given by square summable sequence, then an asymptotically stable and a square summable solution of a deterministic difference equation remains to be an asymptotically mean square stable and a mean square summable solution by stochastic perturbations.     

About stability of difference equations with square summable level of stochastic perturbations

Abstract

This article continues stability investigation of systems with fading stochastic perturbations. In recent results for systems with the continuous time, it was shown that if stochastic perturbations fade on the infinity quickly enough then asymptotically stable deterministic system remains to be an asymptotically mean square stable independently of the magnitude of the intensity maximum of these stochastic perturbations. Here similar statements are obtained for systems with the discrete time by the condition that the level of stochastic perturbations is given by a square summable sequence. Besides the unsolved problem is proposed: is it possible to get analogous results with not so quickly fading stochastic perturbations. This problem is an open problem and for systems with the continuous time too.     

Global behavior of a three-dimensional linear fractional system of difference equations

We investigate the global asymptotic behavior of solutions of the system of difference equations

where the parameters a, b, c, d, e, and f are in (0,∞) and the initial conditions x₀, y₀, and z₀ are arbitrary non-negative numbers. We obtain some global attractivity results for the positive equilibrium of this system for different values of the parameters.     

Some problems on the global attractivity of linear nonautonomous difference equations

This paper is to solve several problems on the global attractivity of the zero solution of the nonautonomous difference equation xn+1 - xn + Pnxn-kn = 0,n ∈ Z(0), where {Pn} is a sequence of nonnegative real numbers, and {kn} is a sequence of nonnegative integers with n - kn→∞ as n→∞.     

About stability of delay differential equations with square integrable level of stochastic perturbations

The long term behavior of solutions of stochastic delay differential equations with a fading stochastic perturbations is investigated. It is shown that if the level of stochastic perturbations fades on the infinity, for instance, if it is given by square integrable function, then an asymptotically stable deterministic system remains to be an asymptotically stable (in mean square).     

Stabilization of two cycles of difference equations with stochastic perturbations

A map which experiences a period doubling route to chaos, under a stochastic perturbation with a positive mean, can have a stable blurred two-cycle for large enough values of the parameter. The limit dynamics of this cycle is described, and it is demonstrated that most well-known population dynamics models (e.g. Ricker, truncated logistic, Hassel and May, Bellows maps) have this stable blurred two-cycle. For a general type of maps, in addition, there may be a blurred stable area near the equilibrium.     

正负系数中立型差分方程的全局吸引性

A neutral difference equation with positive and negative coefficients△(xn-cnxn-k) pnxn-l-qnxn-r=0,n=0,1,2.…，is considered and a sufficient condition for the global attractivity of the zero solution of this equation is obtained, which improves and extends the all known results in the literature.     

On globally attracting two-cycles of second order difference equations

We establish necessary and sufficient conditions so that all solutions of a second order autonomous difference equation are attracted to period-2 solutions. In addition, we prove that difference equations enjoying this property posses invariants or first integrals.     

Estimate of solutions for differential and difference functional equations with applications to difference methods

The theorems on the estimate of solutions for nonlinear second-order partial differential functional equations mainly of parabolic type with Dirichlet’s condition and for the suitable explicit finite difference functional schemes are proved. The proofs are based on the comparison technique. The convergent difference method given is considered without an assumption of the global generalized Perron condition on the functional variable but with local one in some sense only. It is a consequence of our estimate theorems. The functional dependence is of the Volterra type.     

On oscillatory third order difference equations

In this paper, sufficient conditions have been obtained for oscillation/nonoscillation of a class of homogeneous/nonhomogeneous linear difference equations of third order.     

具有可变时滞的非线性非自治差分方程的线性化全局渐近稳定性及其应用

证明了在一定条件下,具有可变时滞的非线性非自治差分方程的全局渐近稳定性可由某种线性差分方程的渐近稳定性确定,给出了这类差分方程全局渐近稳定的充分条件.作为实例,获得了具有可变时滞的离散型非自治广义Log istic方程的全局吸收性判别准则.     

Periodic pattern repetition for unbounded solutions to finite difference equations

ABSTRACT

We study a large class of finite difference equations that exhibit a type of periodic pattern repetition in their solutions for certain choices of initial conditions and prove the existence of unbounded solutions.     

Singular perturbations of abstract wave equations

Given, on the Hilbert space H₀, the self-adjoint operator B and the skew-adjoint operators C₁ and C₂, we consider, on the Hilbert space H?D(B)⊕H₀, the skew-adjoint operator

     

On ordinary difference equations with variable coefficients

A solution representation in matrix form is achieved for all orders of the title equation. Sufficient criteria for the asymptotical stability and the bounded input-bounded output stability of the solution sequence are presented. Our solution representation is confronted with a fully explicit representation. The latter is derived in a simple way and also generalized to a class of partial difference equation with variable coefficients.     

Periodicity and solutions of some rational difference equations systems

In this paper we are interested in a technique for solving some nonlinear rational systems of difference equations of third order, in three-dimensional case. Moreover, we study the periodicity of solutions for such systems. Finally, some numerical examples are presented.