---
tag: 4SH2ZB984E
kind: concept
name: Global attractor
msc: [35B41, 37L25, 37L30]
sources:
  - url: https://en.wikipedia.org/wiki/Attractor
---

## Summary

The global attractor of a dissipative dynamical system on a metric space is a compact invariant set.
It attracts every bounded set of initial data.
It contains all the long-time dynamics: all equilibria, periodic orbits and bounded complete trajectories.
Many dissipative PDEs have a global attractor of finite fractal dimension, although the phase space is infinite-dimensional.
Examples are reaction–diffusion equations and the two-dimensional [[Navier–Stokes equations|18KXZHVF4M]].

## Explanations

### L1 human

Start a dissipative system from any state and wait.
Friction or diffusion wears down most of the details, and the state comes closer and closer to a certain set, the global attractor.
For $u' = u - u^3$ on the real line, every solution tends to $-1$, $0$ or $1$.
The global attractor is the interval $[-1, 1]$, made of the three equilibria and the orbits joining them.
For a dissipative PDE, the attractor is often a finite-dimensional object living inside an infinite-dimensional space.