The Poincare-Bendixson Theorem

The Poincare-Bendixson theorem is the fundamental existence result for periodic orbits in two-dimensional autonomous systems.

Statement

Theorem6.5Poincare-Bendixson theorem (precise)

Consider the autonomous system $\mathbf{x}' = \mathbf{F}(\mathbf{x})$ in $\mathbb{R}^2$ with $\mathbf{F} \in C^1$ . Let $\gamma^+$ be a positive semi-orbit (trajectory for $t \geq 0$ ) that remains in a compact set $K \subset \mathbb{R}^2$ . Then the $\omega$ -limit set $\omega(\gamma^+)$ is one of:

An equilibrium point.
A periodic orbit.
A set consisting of equilibrium points and trajectories connecting them (a heteroclinic or homoclinic cycle).

RemarkCorollary for limit cycles

If the compact set $K$ contains no equilibrium points, then $\omega(\gamma^+)$ must be a periodic orbit. This provides the practical criterion: construct a positively invariant region free of equilibria.

Proof Ideas

Proof

The proof relies on two key lemmas specific to $\mathbb{R}^2$ :

Lemma 1 (Jordan Curve Theorem): A simple closed curve in $\mathbb{R}^2$ separates the plane into interior and exterior.

Lemma 2: If $\Sigma$ is a transversal (line segment crossing trajectories), any trajectory can cross $\Sigma$ only in a monotone fashion (always in the same direction, with successive crossings moving monotonically along $\Sigma$ ).

Using these: if $\gamma^+$ stays in $K$ and meets a transversal $\Sigma$ at points $p_1, p_2, \ldots$ , the sequence is monotone on $\Sigma$ and bounded, hence convergent. The limit determines a periodic orbit in $\omega(\gamma^+)$ .

The full proof shows $\omega(\gamma^+)$ is nonempty (by compactness), closed, connected, and invariant. In the absence of equilibria, it must be a periodic orbit. $\blacksquare$

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Applications

ExampleLotka-Volterra predator-prey

The system $x' = x(\alpha - \beta y)$ , $y' = y(-\gamma + \delta x)$ with $\alpha, \beta, \gamma, \delta > 0$ has a center at $(\gamma/\delta, \alpha/\beta)$ (neutrally stable periodic orbits). Adding logistic growth $x' = x(\alpha - \beta y - \varepsilon x)$ with small $\varepsilon > 0$ creates an unstable equilibrium surrounded by a trapping region, yielding a limit cycle.

RemarkFailure in higher dimensions

The Poincare-Bendixson theorem fails in $\mathbb{R}^3$ and higher. The Lorenz system $\dot{x} = \sigma(y-x)$ , $\dot{y} = rx-y-xz$ , $\dot{z} = xy-bz$ has bounded trajectories that are neither periodic nor converging to equilibria — they exhibit chaos. The Jordan Curve Theorem, essential to the proof, has no analogue in $\mathbb{R}^3$ .