Chaos Lab

Discrete map · 1D · dim 1

Chebyshev map

The k-th Chebyshev polynomial T_k applied iteratively. Chaotic on [−1, 1] for k ≥ 2; semigroup composition T_a ∘ T_b = T_{ab} makes Chebyshev maps appealing for chaos-based cryptography (and famously broken claims of public-key schemes).

Chebyshev T₃ — already fully chaotic on [−1, 1]; λ = ln 3.

Equations

x_{n+1} = cos(k · arccos x_n) = T_k(x_n)

At a glance

Parametersinteger k ≥ 2
Chaotic foralways (for k ≥ 2)
Lyapunov exponentλ = ln k
HistoryUsed in public-key chaos crypto by Kocarev–Tasev (2003), broken by Bergamo et al. (2005).

Try it

Open the interactive playground at /tools/cobweb.

See also

Quick quiz

Test yourself on chebyshev

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

  1. Q1.Chebyshev maps T_n satisfy the composition rule:

  2. Q2.T_3(x) equals:

  3. Q3.The Lyapunov exponent of iterated T_k is:

  4. Q4.T_n(cos θ) equals:

  5. Q5.Public-key chaos crypto from Chebyshev maps (Kocarev-Tasev 2003) was:

  6. Q6.Chebyshev T_n is everywhere chaotic on:

  7. Q7.What is the invariant density of T_n on [−1, 1]?

  8. Q8.T_2(x) =

0 of 8 answered