Attractors: persistence, and density of their basins

Abstract

An investigation of qualitative features of flows on manifolds, in terms of their attractors and quasi-attractors.A quasi-attractor is any nonempty intersection of attractors.It is shown that quasi-attractors other than attractors occur for a large set of flows.It is also shown that for a generic flow (for each flow in a residual subset of the set of all flows), each attractor "persists" as an attractor of all nearby flows.Similar statements are shown to hold with "quasi-attractor", "chain transitive attractor", and "chain transitive quasi-attractor" in place of "attractor".Finally, the set of flows under which almost all points tend asymptotically to a chain transitive quasi-attractor is characterized in terms of stable sets of invariant sets.