---
tag: 0EW1X61ZT1
kind: statement
role: theorem
name: Poincaré–Bendixson theorem
msc: [34C05, 37C27]
uses: [0EERDDPSVT, 01DSNP0B4R, 0G4X36AE0X, 02903SSA48]
sources:
  - url: https://en.wikipedia.org/wiki/Poincar%C3%A9%E2%80%93Bendixson_theorem
    locator: Theorem
same_as:
  wikidata: Q248931
  wikipedia: Poincaré–Bendixson theorem
---

## Statement

Let $U$ be an open subset of $\mathbb{R}^2$, let $f : U \to \mathbb{R}^2$ be continuously differentiable, and let $\varphi$ be the [[flow|02903SSA48]] of $x' = f(x)$.
Let $y \in U$ be a point whose solution exists for all $t \ge 0$. Let $\omega(y)$ be its $\omega$-limit set, as in the definition of a [[limit cycle|0EERDDPSVT]].
Assume that $\omega(y)$ is non-empty and compact, and that it contains only finitely many [[equilibrium points|0G4X36AE0X]].
Then $\omega(y)$ is one of the following three kinds of set.
(1) It is a single equilibrium point.
(2) It is a [[periodic orbit|01DSNP0B4R]].
(3) It is a connected set of finitely many equilibrium points and of trajectories that tend to these points as $t \to \infty$ and as $t \to -\infty$.

## Remarks

A common corollary: let $K \subseteq U$ be compact, contain no equilibrium point, and contain the positive half-trajectory of $y$. Then $\omega(y)$ is a periodic orbit.
To apply it, one often finds an annulus that trajectories enter but cannot leave. Then the annulus contains a periodic orbit.
The theorem is special to the plane. It fails in $\mathbb{R}^3$, where chaotic behaviour is possible, as for the Lorenz system.
The trajectories in case (3) are homoclinic or heteroclinic orbits.

## History

The theorem is named after Henri Poincaré and Ivar Bendixson.

## Explanations

### L1 human

In the plane, a bounded motion has few options in the long run.
It settles at an equilibrium, approaches a closed loop, or follows a chain of paths between equilibria.
Trajectories in the plane cannot cross, and this is what limits the possibilities.