Forward attractors and limit sets of nonautonomous difference equations

The theory of nonautonomous dynamical systems has undergone major development during

the past 19 years since I talked about attractors of nonautonomous difference equations at ICDEA Poznan in 1998.

Two types of attractors consisting of invariant families of sets 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 behavior.

The forward asymptotic behavior can also be described through the omega-limit set of the system. This set is closely related to what Vishik called the uniform attractor although it need not be invariant. It is shown to be asymptotically positively invariant and also, provided a future uniformity condition holds, also asymptotically positively invariant. Hence this omega-limit set provides useful information about the behavior in current time during the approach to the future limit.