Discrete map · 1D · dim 1
Logistic map
Robert May's 1976 review introduced this one-parameter quadratic family as the canonical example of complex dynamics from simple rules. It is the type-specimen of the period-doubling route to chaos and the source of the Feigenbaum constants.
Equations
x_{n+1} = r · x_n · (1 − x_n)At a glance
| Parameters | r ∈ [0, 4] |
|---|---|
| Chaotic for | r ≈ 3.5699 ≤ r ≤ 4 (chaotic windows and periodic islands) |
| Lyapunov exponent | λ(r = 4) = ln 2 ≈ 0.6931 |
| History | May (1976), Feigenbaum (1978). |
Try it
Open the interactive playground at /tools/logistic.
See also
Quick quiz
Test yourself on logistic
8 multiple-choice questions. Pick an answer for each, then submit to see explanations.
Q1.The logistic map x_{n+1} = r·x_n·(1 − x_n) becomes chaotic for r above:
Q2.Lyapunov exponent of the logistic map at r = 4 equals:
Q3.The Feigenbaum constant δ is approximately:
Q4.Between r = 3 and r ≈ 3.4495 the orbit is:
Q5.Robert May popularised this map in:
Q6.A famous period-3 window is centred near:
Q7.Logistic, sine, and tent maps all share the universal:
Q8.What does r = 1 mark?