Ideas behind the proof: Consider a trajectory that starts on the positive x-axis near the origin (Figure 6.6.2). Sufficiently near the origin, the flow swirls around the origin, thanks to the dominant influence of the linear center, and so the trajectory eventually intersects the negative x-axis. (This is the step where our proof lacks rigor, but the claim should seem plausible.) Now we use reversibility. By reflecting the trajectory across the x-axis, and changing the sign of t, we obtain a twin trajectory with the same endpoints but with its arrow reversed (Figure 6.6.3). Together the two trajectories form a closed orbit, as desired. Hence all trajectories sufficiently close to the origin are closed.