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 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. Assume that $\omega(y)$ is non-empty and compact, and that it contains only finitely many equilibrium points. 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. (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.