Global Attractor

A global attractor for a continuous semiflow on a metric space is a nonempty compact set that is strictly invariant, meaning for , and attracts every bounded set in . More explicitly, for every bounded set ,

Here is the metric, and the semiflow satisfies and for (Temam 1997).

A global attractor need not contain a dense trajectory. It can contain several equilibrium points, periodic orbits, and trajectories joining them. For example, the differential equation on has global attractor , containing the three equilibrium points , 0, and 1. This broader notion is relevant to the unrestricted formulation of Eden's conjecture.