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.     

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.     

Stability of certain nonautonomous difference equations

In this paper, the stability of a class of nonautonomous nonlinear difference equations that are determined by functions which satisfy certain contractive type condition is studied in metric spaces, and its variant problem in ordered Banach spaces is also investigated. Several examples are given to illustrate their applications.     

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.     

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.     

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.     

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.     

Almost automorphic solutions of nonautonomous evolution equations

In this paper, we establish some new theorems about the existence of almost automorphic solutions to nonautonomous evolution equations u^′(t)=A(t)u(t)+f(t) and u^′(t)=A(t)u(t)+f(t,u(t)) in Banach spaces. As we will see, our results allow for a more general A(t) to some extent. An example is also given to illustrate our results. In addition, by means of an example, we show that one cannot ensure the existence of almost automorphic solutions to u^′(t)=A(t)u(t)+f(t) even if the evolution family U(t,s) generated by A(t) is exponentially stable and fAA(X).     

Oscillations of the system of linear nonautonomous difference equations

We consider the linear nonautonomous system of difference equations x_n+1−x_n+P(n)x_n−k=0, n=0,1,2,… , where k∈Z, P(n)∈R^rxr. We obtain sufficient conditions for the system to be oscillatory. The conditions based on the eigenvalues of the matrix coefficients of the system.     

Stability for solutions to nonlinear nonautonomous evolution equations

In this paper, we find an estimate on d(u(t), K(t)), where u is a mild solution to the nonautonomous Cauchy problem \({\dot{u}(t) + A(t)u(t) \ni 0,\, t \geq s, u(s) = u_0}\) . Here, A(t) is a family of nonlinear multivalued, ω-accretive operators in a Banach space X, with D(A(t)) possibly depending on t, and K(t) a family of closed subsets in X.     

Taylor approximation of invariant fiber bundles for nonautonomous difference equations

Invariant fiber bundles generalize invariant manifolds to nonautonomous difference equations. In this paper we develop a method to calculate their Taylor approximation, which is of crucial importance, e.g. for an application of the reduction principle in a nonautonomous setting.     

Pseudo-almost periodicity of some nonautonomous evolution equations with delay

This paper is concerned with pseudo-almost periodicity of the solutions to the nonautonomous evolution equation with delay u^′(t)=A(t)u(t)+f(t,u(t−h))$u^{\prime }\left(t\right)=A\left(t\right)u\left(t\right)+f(t,u(t-h))$. Some sufficient conditions which ensure the existence and uniqueness of pseudo-almost periodic mild solutions to the evolution equation with delay are given. An example is shown to illustrate our results.     

Solving a class of nonautonomous difference equations by generalized invariants

By using generalized invariants, we describe a method for solving a class of higher‐order nonautonomous difference equations. Solvability of the equations of order two, three, and four are studied in detail. Our results explain and extend some problems in the literature. As far as we know, the case when the order is four is considered for the first time in this paper. For the equations of second order, we also give an explanation how they can be obtained in a natural way.     

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.