The Bifurcation Diagram - Route to Chaos

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Resolution (samples per r)

200

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r range [2.8, 4.0]

x range [0, 1]

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The Logistic Map

x_n+1 = r · x_n · (1 - x_n)

Key Bifurcation Points

Period 1 → 2 r = 3.0

Period 2 → 4 r ≈ 3.449

Period 4 → 8 r ≈ 3.544

Onset of chaos r ≈ 3.5699

Period-3 window r ≈ 3.8284

Feigenbaum δ 4.6692...

🗺️ Reading the Diagram

The horizontal axis is the growth rate r. The vertical axis shows the long-term values of x after many iterations. One dot = stable fixed point. Two dots = period-2 cycle. The "dust" region = chaos (infinitely many values visited).

🔀 Period Doubling

As r increases, stable behavior splits: 1 → 2 → 4 → 8 → 16... The gaps between bifurcations shrink by the Feigenbaum constant δ ≈ 4.669. This universal ratio appears in all systems with period-doubling routes to chaos!

🌊 Windows of Order

Within the chaotic region, islands of stability appear. At r ≈ 3.83, a stable period-3 cycle emerges, then period-doubles itself! "Period 3 implies chaos" — Yorke & Li's famous theorem. Order and chaos interweave.

🔬 Self-Similarity

Zoom into any branch of the diagram and you'll find smaller copies of the whole. This fractal structure reveals deep connections between chaos and geometry. The bifurcation diagram is itself a kind of fractal!