The Logistic Map
One of the most famous equations in all of chaos theory — and it's just multiplication. A population is x (0 = none, 1 = the most possible). Each year the next value is x → r · x · (1 − x). Turn the knob r and watch what a single rule can do.
The knob
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Period-doubling into chaos
r below 3: the value settles to one steady number. Totally predictable.
r ≈ 3 to 3.45: it splits — bouncing between two values forever (period-2). Then four. Then eight…
r above ≈ 3.57: the splits pile up so fast the value never repeats. That's chaos — and it's hiding inside multiplication.
The diagram is a map of predictability
Each split in the bifurcation diagram is a doubling of how many futures are possible. The solid lines are predictable; the dark fuzzy bands are chaos.
Look closely inside the chaos and you'll spot clear vertical gaps — windows of order where predictability briefly returns. Try r ≈ 3.83. Order and chaos are tangled together.
For teachers & grown-ups
The "Over time" view plots the orbit xₙ versus iteration n; the "Cobweb" view shows the same iteration geometrically against the parabola y = r·x·(1−x) and the line y = x, so students can see why the value lands where it does. The bifurcation diagram is built by iterating each r value hundreds of times, discarding the transient, and plotting the long-run values — the points it settles onto. The period-doubling cascade (period 1 → 2 → 4 → 8 → …) accumulates at r ≈ 3.5699, after which chaos begins; the spacing of those splits shrinks by a near-constant factor (Feigenbaum's constant, ≈ 4.669). The "windows of order" — e.g. the period-3 window near r ≈ 3.83 — are a great hook: chaos is not the same as randomness, and order can re-emerge. Changing x₀ in the chaotic regime gives a totally different orbit (sensitive dependence) but the same bifurcation picture, a nice distinction between a single trajectory and the attractor.