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The Poincare-Bendixson theorem states that for a continuous dynamical system in the plane, a trajectory confined to a bounded region that contains no fixed points must approach a periodic orbit. This theorem is simultaneously a powerful existence result for limit cycles and a topological impossibility result for chaos: it tells you that two-dimensional continuous flows are "too simple" for chaos. Chaotic behavior requires at least three continuous dimensions.
The Poincare-Bendixson theorem is one of the deepest results in two-dimensional dynamics, and it works in two directions simultaneously. First, it is a powerful existence theorem: it lets you prove that a limit cycle exists without ever finding or solving for it. Second, it is an impossibility theorem: it proves that chaos cannot occur in two-dimensional continuous flows. Both consequences flow from the topology of the plane.
The theorem's statement is elegant. Consider a continuous dynamical system in the plane. If a trajectory is confined to a bounded region for all future time, then its omega-limit set (the set of points it accumulates on as t → ∞) must be one of three things: a fixed point, a periodic orbit, or a finite union of fixed points connected by trajectories (a heteroclinic cycle). That's it. No other long-term behaviors are possible in two continuous dimensions. If you can additionally rule out fixed points in the trapping region, only periodic orbits remain.
The practical application is a two-step recipe for proving limit cycles exist. Step one: find a trapping region — a bounded region of phase space that the flow cannot leave. This usually means showing the vector field points inward on the boundary. Step two: show the trapping region contains no stable fixed points. If there are no fixed points at all, you're done — Poincare-Bendixson guarantees a periodic orbit. If there are unstable fixed points, construct an annular trapping region with the unstable fixed point excised from the interior. This strategy was used by van der Pol and Lienard to prove limit cycles exist in nonlinear oscillators, and it remains a standard tool.
The impossibility of chaos in 2D continuous systems is the theorem's most profound consequence. Chaos requires sensitive dependence on initial conditions — nearby trajectories must diverge exponentially. But in two dimensions, the no-crossing property of continuous flows prevents trajectories from mixing in the complicated way chaos requires. A trajectory in 2D divides the plane into two regions, and other trajectories can't cross from one side to the other. This topological constraint is so restrictive that the only possible long-term behaviors are fixed points and periodic orbits. In three dimensions, trajectories can weave over and under each other, and the Poincare-Bendixson constraint evaporates. This is precisely why the simplest chaotic continuous systems — the Lorenz system, the Rossler system — live in three dimensions.