---
tag: 681NY7506V
kind: definition
name: Global attractor
msc: [35B41, 37L25, 37L30]
defines: 4SH2ZB984E
uses: [0VKD1TP7F3, 00R0X1WGPX, WWZTW11664]
sources:
  - url: https://en.wikipedia.org/wiki/Attractor
---

## Definition

Let $(X, d)$ be a [[metric space|0VKD1TP7F3]] and let $(S(t))_{t \ge 0}$ be a semigroup of continuous maps $S(t) \colon X \to X$, with $S(0) = \id$ and $S(t + s) = S(t) S(s)$.
For subsets $A, B \subseteq X$, let $\operatorname{dist}(B, A) = \sup_{b \in B} \inf_{a \in A} d(a, b)$.
A *global attractor* of the semigroup is a set $\mathcal{A} \subseteq X$ such that:

1. $\mathcal{A}$ is [[compact|00R0X1WGPX]];
2. $\mathcal{A}$ is [[invariant|WWZTW11664]]: $S(t) \mathcal{A} = \mathcal{A}$ for every $t \ge 0$;
3. $\mathcal{A}$ attracts bounded sets: $\operatorname{dist}(S(t) B, \mathcal{A}) \to 0$ as $t \to \infty$ for every bounded set $B \subseteq X$.

## Remarks

A global attractor, when it exists, is unique: it is the largest compact invariant set and the smallest closed set that attracts bounded sets.

Example: let $X = \R$ and let $S(t)$ be the flow of $u' = u - u^3$.
Every solution with $u(0) > 0$ tends to $1$, every solution with $u(0) < 0$ tends to $-1$, and $0$ is fixed.
The set $\mathcal{A} = [-1, 1]$ is compact, invariant since the flow maps $[-1, 1]$ onto itself, and attracts every bounded interval.

Non-example: for the same flow, the set $\set{-1, 1}$ is compact and invariant, but it does not attract the bounded set $\set{0}$, since $S(t) 0 = 0$ for all $t$.
So clause 3 fails.

A standard sufficient condition is that the semigroup has a bounded absorbing set and is asymptotically compact.