On the forward dynamical behavior of nonautonomous systems

This paper is concerned with the forward dynamical behavior of nonautonomous systems. Under some general conditions, it is shown that in an arbitrary small neighborhood of a pullback attractor of a nonautonomous system, there exists a family of sets $ \{\mathcal{A}_\varepsilon(p)\}_{p\in P} $ of phase space $ X $, which is forward invariant such that $ \{\mathcal {A}_\varepsilon(p)\}_{p\in P} $ uniformly forward attracts each bounded subset of $ X $. Furthermore, we can also prove that $ \{\mathcal{A}_\varepsilon(p)\}_{p\in P} $ forward attracts each bounded set at an exponential rate.