Tool
Logistic map playground
The canonical example of how a single nonlinear equation can produce the entire period-doubling cascade. Slide r from 3 toward 4 and watch the orbit go from fixed point to 2-cycle to chaos.
Time series (x_n vs n)
Cobweb plot
Iteration rule
x_{n+1} = r · x_n · (1 − x_n)Numerical Lyapunov estimate at these parameters: λ ≈ 0.3550: chaotic (orbit diverges from neighbours).
Landmarks on r
- r < 1: extinction (orbit → 0)
- 1 < r < 3: stable fixed point
- r = 3 to ≈3.4495: period 2
- ≈3.4495 to ≈3.5441: period 4
- ≈3.5441 to ≈3.5644: period 8 (and on by 2)
- r ≈ 3.56995: Feigenbaum point, onset of chaos
- 3.5699 < r < 4: chaos with periodic windows (period-3 at r ≈ 3.8284)
- r = 4: full chaos, λ = ln 2 ≈ 0.6931
Background at /maps/logistic and the bifurcation studio at /tools/bifurcation.
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