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).
Equations
x_{n+1} = cos(k · arccos x_n) = T_k(x_n)At a glance
| Parameters | integer k ≥ 2 |
|---|---|
| Chaotic for | always (for k ≥ 2) |
| Lyapunov exponent | λ = ln k |
| History | Used in public-key chaos crypto by Kocarev–Tasev (2003), broken by Bergamo et al. (2005). |
Try it
Open the interactive playground at /tools/cobweb.