Topological simplification of nonautonomous difference equations

In this paper, we continue the study of geometric properties of nonautonomous difference equations in arbitrary Banach spaces which was begun in [2 Aulbach, B. 1998. The fundamental existence theorem on invariant fiber bundles. Journal of Difference Equations and Applications, 3(5–6): 501–537.  [Google Scholar],3 Aulbach, B. and Wanner, T. 2003. Invariant foliations and decoupling of non-autonomous difference equations. Journal of Difference Equations and Applications, 9(5): 459–472. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]]. Building on previous results on invariant fiber bundles and foliations, this paper addresses the problem of topological simplifications via continuous conjugacies and semiconjugacies. In particular, we establish a reduction principle for not necessarily invertible difference equations, as well as a generalized Hartman–Grobman theorem for systems with not necessarily invertible linear part.     

Stability of nonautonomous difference equations with several delays

We consider the possibility to construct efficient stability criteria for solutions to difference equations with variable coefficients. We prove that one can associate a difference equation with a certain functional differential equation, whose solution has the same asymptotic behavior. We adduce examples, demonstrating the essential character of conditions of the obtained theorems and the exactness of the constant 3/2 which defines the boundary of the stability domain.     

Global attractivity of a family of nonautonomous max-type difference equations

This paper studies a family of nonautonomous max-type difference equations with several delays. Using some transformations we prove global attractivity of positive solutions to these equations under some conditions. Some of our results considerably extend related ones in the literature. Two examples are given to illustrate the main results.     

A reduction technique for discrete generalized algebraic and difference Riccati equations

This paper proposes a reduction technique for the generalized Riccati difference equation arising in optimal control and optimal filtering. This technique relies on a study on the generalized discrete algebraic Riccati equation. In particular, an analysis on the eigenstructure of the corresponding extended symplectic pencil enables to identify a subspace in which all the solutions of the generalized discrete algebraic Riccati equation are coincident. This subspace is the key to derive a decomposition technique for the generalized Riccati difference equation. This decomposition isolates a “nilpotent” part, which converges to a steady-state solution in a finite number of steps, from another part that can be computed by iterating a reduced-order generalized Riccati difference equation.     

Regular and extremal solutions for difference equations with generalized phi-Laplacian

Non-oscillatory solutions for second-order difference equations with generalized phi-Laplacian are studied. Solutions are classified according to the asymptotic behaviour as regular or extremal solutions. Their existence and possible coexistence are investigated as well. In particular, the existence of infinitely many extremal solutions for equations with the discrete mean curvature operator is proved by means of an iterative method. This paper is completed by examples and some open problems.     

Towards a bifurcation theory for nonautonomous difference equations

Although, bifurcation theory of equations with autonomous and periodic time dependence is a major object of research in the study of dynamical systems since decades, the notion of a nonautonomous bifurcation is not yet established. This article contains an approach to overcome this deficit in the context of nonautonomous difference equations. Based on special notions of attractivity and repulsivity, nonautonomous bifurcation phenomena are studied. We obtain generalizations of the well-known one-dimensional transcritical and pitchfork bifurcation.     

Forward attraction in nonautonomous difference equations

Two types of attractors consisting of families of sets that are mapped into each other under the dynamics have been defined for nonautonomous difference equations, one using pullback convergence with information about the system in the past and the other using forward convergence with information about the system in the future. In both cases, the component sets are constructed using a pullback argument within a positively invariant family of sets. The forward attractor so constructed also uses information about the past, which is very restrictive and not essential for determining future behaviour. Here an alternative is investigated, essentially the omega-limit set of the system, which Chepyzhov and Vishik called the uniform attractor. It is shown here that this set is asymptotically positively invariant, thus providing it with an hitherto missing form of invariance, if in somewhat weaker than usual, that one expects an attractor to possess. As a consequence this set provides useful information about the behaviour in current time during the approach to the limit.     

Asymptotic behavior of solutions for a class of retarded difference equations

Consider the retarded difference equationx _n −x _n−1 =F(−f(x _n )+g(x _n−k )), wherek is a positive integer,F,f,g:R→R are continuous,F andf are increasing onR, anduF(u)>0 for allu≠0. We show that whenf(y)≥g(y) (resp. f(y)≤g(y)) fory∈R, every solution of (*) tends to either a constant or −∞ (resp. ∞) asn→∞. Furthermore, iff(y)≡g(y) fory∈R, then every solution of (*) tends to a constant asn→∞. Project supported by NNSF (19601016) of China and NSF (97-37-42) of Hunan     

Pullback attractors in nonautonomous difference equations

Nonautonomous difference equations are formulated as cocycles which generalize semigroups corresponding to autonomous difference equations. Pullback attractors are the appropriate generalization of autonomous attractors to cocycles. The existence of a pullback attractor follows when the difference equation cocycle has a pullback absorbing set. Results from the literature are outlined, including the construction of a Lyapunov function characterizing pullback attraction, and illustrated with several examples.     

On the behaviour of nonautonomous difference equations

We obtain in this work a repeller version of a criterion previously obtained for the exponential stability and improve a condition for asymptotic stability of equilibrium points of the nonautonomous higher order difference equations by weak contraction arguments.     

On the stability of nonautonomous difference equations

In this paper, we obtain new sufficient conditions for the asymptotic stability and instability of equilibrium points of the nonautonomous higher order difference equations by means of weak contractions and weak expansions.     

On a class of systems of rational second-order difference equations solvable in closed form

We show that the following nonlinear system of difference equations where parameters a,b,c,d and initial values x₋₁,x₀,y₋₁,y₀ are real numbers, is solvable in closed form, considerably generalizing some recent results. To do this, we use the method of transformation along with several tricks, transforming the system to some known solvable difference equations, by use of which we obtain some closed-form formulas for general solution to the system. The following five cases are considered separately: (1) c=0; (2) d=0; (3) a=0; (4) b=0; and (5) abcd≠0.     

New global stability conditions for a class of difference equations

In this paper we consider some classes of difference equations, including the well-known Clark model, and study the stability of their solutions. In order to do that we introduce a property, namely semicontractivity, and study relations between ‘semi-contractive’ functions and sufficient conditions for the solution of the difference equation to be globally asymptotically stable. Moreover, we establish new sufficient conditions for the solution to be globally asymptotically stable, and we improve the ‘3/2 criteria’ type stability conditions.      

SOLITON SOLUTIONS OF A CLASS OF GENERALIZED NONLINEAR SCHRODINGER EQUATIONS

Soliton solutions of a class of generalized nonlinear evolution equations are discussed ana-lytically and numerically. This is done by using a travelling wave method to formulate one-soliton solution and the finite difference method to the numerical solutions and the interactions betweenthe solitons for the generalized nonlinear Sehrodinger equations. the characteristic behavior of thenonlinearity admintted in the system has been investigated and the soliton states of the system in thelimit when a→Oand a→∞ have been studled. The results presented show that the soliton phe-rtomenon is charaeteristics associated with the nonlinearities of the dynamical systems.     

具连续变量的偶数阶中立型差分方程的振动性

研究具有连续变量的偶数阶中立型时滞差分方程的解的振动性,给出了有界解振动的几个充分条件.