---
tag: 4V09M20SJB
kind: definition
name: Limit set
msc: [37C10, 34C05]
defines: YF4XH2BEG5
uses: [07BNRFQ0PJ, 02903SSA48, 0XKWSSY8P2]
sources:
  - url: https://en.wikipedia.org/wiki/Limit_set
---

## Definition

Let $U$ be an open subset of $\R^n$, let $f \colon U \to \R^n$ be locally Lipschitz, and let $\varphi$ be the [[flow|02903SSA48]] of the [[autonomous system|07BNRFQ0PJ]] $x' = f(x)$.
Let $x \in U$ be such that the solution through $x$ is defined for all $t \ge 0$.
The $\omega$-limit set of $x$ is the set $\omega(x)$ of points $y \in U$ for which there is a sequence $t_k \to +\infty$ with $\varphi_{t_k}(x) \to y$.
Let the solution through $x$ be defined for all $t \le 0$.
The $\alpha$-limit set $\alpha(x)$ is the set of points $y \in U$ with $\varphi_{t_k}(x) \to y$ for some sequence $t_k \to -\infty$.

## Remarks

Equivalently, $\omega(x) = \bigcap_{T \ge 0} \overline{\set{\varphi_t(x) : t \ge T}} \cap U$, where the bar denotes the closure in $\R^n$.
The set $\omega(x)$ is closed in $U$ and [[invariant|WWZTW11664]].
If the forward orbit of $x$ lies in a compact subset of $U$, then $\omega(x)$ is nonempty, compact and connected, and $\varphi_t(x)$ tends to $\omega(x)$ as $t \to \infty$.
Example: in polar coordinates, let $r' = r(1 - r^2)$ and $\theta' = 1$ in the plane.
For $x \ne 0$ the radius tends to $1$ while the angle keeps turning.
So every point of the unit circle is a limit of $\varphi_{t_k}(x)$ for suitable $t_k \to \infty$, and $\omega(x)$ is the unit circle.
For $0 < \abs{x} < 1$, the radius tends to $0$ as $t \to -\infty$, so $\alpha(x) = \set{0}$.
Non-example: for $x' = 1$ on $\R$, the solution $\varphi_t(x) = x + t$ has no convergent sequence $\varphi_{t_k}(x)$ with $t_k \to \infty$, so $\omega(x)$ is empty.
This is possible because the forward orbit is not bounded.

## Explanations

### L1 human

The $\omega$-limit set collects all points that the forward motion comes close to again and again.
The $\alpha$-limit set does the same for the motion run backwards in time.