Period Doubling

The road to chaos

An interactive lesson with 6 steps, about 5 minutes: you predict, drag and solve, with instant feedback on every answer. It runs in your browser and needs JavaScript.

What you’ll learn

• A steady state is stable while the slope there lies between −1 and 1; for the logistic map that fails at r = 3.
• Period doubling (2, 4, 8, …) leads to chaos at r ≈ 3.57, with calm windows inside.
• Period three implies chaos (Li and Yorke, 1975).
• The doublings close in by Feigenbaum’s universal ratio, δ ≈ 4.669.

Steps

1. Cobwebs
2. When does it wobble?
3. The bifurcation diagram
4. Period three implies chaos
5. The edge of chaos
6. Feigenbaum’s constant