We provide a conceptual outline of the proof of the Poincare-Bendixson theorem, focusing on the key ideas rather than technical details.

Theorem Statement (Simplified Version): Let $\gamma$ be a non-closed trajectory in $\mathbb{R}^2$ that remains in a closed, bounded region $K$ containing no fixed points for all $t \geq 0$. Then the $\omega$-limit set of $\gamma$ is a closed orbit.

Proof Outline:

Step 1: Properties of $\omega$-limit sets. The $\omega$-limit set $\omega(\gamma)$ is the set of accumulation points of $\{\phi_t(\mathbf{x}_0) : t \geq 0\}$. Since $K$ is compact and contains the forward trajectory, $\omega(\gamma)$ is nonempty, compact, and invariant under the flow. By assumption, $\omega(\gamma)$ contains no fixed points.

Step 2: Non-crossing of trajectories. In planar systems, distinct trajectories cannot cross (by uniqueness of solutions to ODEs). This topological constraint is crucial. If two trajectories crossed at a point $p$, there would be two different solutions starting from $p$, contradicting uniqueness.

Step 3: Poincare section and return map. Choose a point $p \in \omega(\gamma)$ and construct a small transverse section $\Sigma$ through $p$—a line segment perpendicular to the vector field. Since $p$ is in the $\omega$-limit set, the trajectory $\gamma$ returns arbitrarily close to $p$ infinitely often, hence intersects $\Sigma$ infinitely many times.

Let $\{p_n\}$ be the sequence of intersection points. Since $\Sigma$ is one-dimensional and bounded, and trajectories cannot cross, the sequence $\{p_n\}$ must be monotonic along $\Sigma$ (always approaching from the same side). Therefore, $\{p_n\}$ converges to some point $q \in \Sigma$.

Step 4: The limit point $q$ lies in $\omega(\gamma)$. By construction, $q = \lim_{n \to \infty} p_n$ is an accumulation point of the trajectory, so $q \in \omega(\gamma)$. Moreover, since $\omega(\gamma)$ is invariant, the entire trajectory through $q$ lies in $\omega(\gamma)$.

Step 5: The trajectory through $q$ is closed. Consider the trajectory $\gamma_q$ starting at $q$. This trajectory must return to $\Sigma$ again at some point $q'$ (since it stays in the compact region $\omega(\gamma)$ and cannot approach a fixed point). By invariance of $\omega(\gamma)$, we have $q' \in \omega(\gamma)$.

The key insight: if $q' \neq q$, then by the monotonicity argument in Step 3, the sequence would continue moving along $\Sigma$, contradicting the fact that $q$ is the limit. Therefore, $q' = q$, meaning the trajectory returns exactly to its starting point. Thus, $\gamma_q$ is a closed orbit.

Step 6: The original trajectory approaches the closed orbit. Having established that $\omega(\gamma)$ contains a closed orbit $\gamma_q$, we must show $\omega(\gamma) = \gamma_q$ (i.e., the limit set consists of exactly this orbit). This follows from analyzing the Poincare map on $\Sigma$: if $\omega(\gamma)$ contained points not on $\gamma_q$, the non-crossing property and compactness would be violated.

Conclusion: Any trajectory confined to a compact region without fixed points must approach a closed orbit. Combined with the case where fixed points are present (in which case trajectories approach fixed points or heteroclinic connections), this gives the complete Poincare-Bendixson theorem.