TheoremProved

Poincaré–Bendixson theorem

Statement

Let $\dot{x} = f(x)$ be a $C^1$ planar vector field, and suppose a forward orbit stays inside a compact region containing no fixed points. Then the $\omega$ -limit set of that orbit is a periodic orbit.

Why is it true?

In two dimensions a trajectory that never leaves a bounded region and never settles at a fixed point cannot wander forever without repeating itself, because it cannot cross itself and is trapped by the topology of the plane; eventually it must spiral onto a closed loop and go around forever. This is a genuinely planar phenomenon — chaos becomes possible only from three dimensions on, once trajectories have room to weave past each other without crossing.

Proof sketch

Show that the $\omega$ -limit set $\omega$ is nonempty, compact, connected and invariant, using compactness of the trapping region. If $\omega$ contains no fixed point, take any point $p \in \omega$ and its orbit; using a transversal segment (a short arc crossing the flow) through $p$ and the Jordan curve theorem, show successive intersections of the orbit through $p$ with the transversal are monotone, which combined with $\omega$ being a single orbit's limit set forces that orbit to be a closed (periodic) curve, and $\omega$ equal to it.

Stated by

stated 1881Henri Poincaré

Topics that use this theorem

Stability of dynamical systems

Related theorems

Birkhoff ergodic theorem

Step-by-step proofs

No step-by-step proof yet for this theorem.

References

Morris W. Hirsch, Stephen Smale, Robert L. Devaney (2004). Differential Equations, Dynamical Systems, and an Introduction to Chaos