Attraction and Lyapunov stability for control systems on vector bundles

Let π:E→Bπ:E→B be a finite-dimensional vector bundle whose base space is compact. In this paper, we study attraction and Lyapunov stability for control systems on EE. We prove that, under certain conditions, the concepts of Conley attractor, uniform attractor, attractor, exponential attractor, asymptotically stable set and stable set are equivalent for the zero section of π:E→Bπ:E→B.