Period doubling

On the left is the bifurcation diagram: above each r, the values the sequence settles on after a long run. Drag r, or press play. In the cobweb on the right the staircase bounces between the curve and the diagonal: below r = 3 it closes in on a single point, the stable fixed point. Past 3 that point turns unstable and the staircase settles into a gold square, alternating between two values: period 2. Further right the sequence visits 4 values before it repeats, then 8, … The bifurcations crowd closer and closer, each gap about δ ≈ 4.669 times the next. When they blur, switch to Zoom 1 or Zoom 2: each level magnifies the diagram δ times across, centred on r∞, and similar forks appear again.