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

Parameters	(no free parameter)
Chaotic for	always
Lyapunov exponent	λ = ln 2
History	Studied since Hadamard; canonical example of Bernoulli shift.

Open the interactive playground at /tools/cobweb.