Abstract
Abstract A Morse decomposition of a global attractor describes its internal dynamics, i.e., the dynamics on invariant compact sets in the attractor and the connections between them. When we deal with non-autonomous dynamical systems, the concept of pullback attractor for the associated skew-product flow appears as a powerful tool to analyze the asymptotic behaviour of these systems. In this paper we develop a Morse decomposition theory for pullback attractors of non-autonomous dynamical systems in Banach spaces with compact base space which, in particular, defines a (non-autonomous) Lyapunov functional on the attractor describing a decaying energy level on the evolution of trajectories.
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