Chaos Lab

Discrete map · 1D · dim 1

Bernoulli (doubling) map

Mod-1 doubling. Acts as a left shift on the binary expansion of x. Chaotic everywhere, with constant Lyapunov exponent ln 2.

All values of x ∈ [0, 1) are visited — Bernoulli is everywhere chaotic with λ = ln 2.

Equations

x_{n+1} = 2 x_n mod 1

At a glance

Parameters(no free parameter)
Chaotic foralways
Lyapunov exponentλ = ln 2
HistoryStudied since Hadamard; canonical example of Bernoulli shift.

Try it

Open the interactive playground at /tools/cobweb.

See also

Quick quiz

Test yourself on bernoulli

8 multiple-choice questions. Pick an answer for each, then submit to see explanations.

  1. Q1.The Bernoulli map x ↦ 2x mod 1 acts on binary expansions as a:

  2. Q2.Its Lyapunov exponent is:

  3. Q3.The Bernoulli map is conjugate to:

  4. Q4.Periodic orbits of the Bernoulli map correspond to:

  5. Q5.The invariant probability measure of the Bernoulli map is:

  6. Q6.Topological entropy of the Bernoulli map is:

  7. Q7.Why is the Bernoulli shift dangerous as a PRNG building block in floating point?

  8. Q8.The Bernoulli shift is an example of which kind of system?

0 of 8 answered