Chebyshev

Systems / Maps / Geometric maps

Iterated Chebyshev polynomials — exactly chaotic maps of the interval.

discrete · map1 dimensionchaotic

return map · bifurcation diagram

Definition

@staticmethod
def _step(X, a):
    x = X
    return np.cos(a * np.arccos(x))

Parameters

Symbol	Default	Role
a	6	map degree (integer a>=2 gives Lyapunov exponent ln a)

Properties

Lyapunov spectrum

$+1.792$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 1$

Divergence ∇·f

n/a — discrete map — flow divergence undefined (per-step contraction is |det J|)

Equilibria

5 fixed points
0 stable · 5 unstable

Define it in TSDynamics

import tsdynamics as ts

sys = ts.systems.Chebyshev()
traj = sys.iterate(steps=10_000)

exps = sys.lyapunov_spectrum()
ts.kaplan_yorke_dimension(exps)

Reference

Adler & Rivlin (1964), Proc. Amer. Math. Soc. 15, 794-796

BibTeX

@misc{chebyshev,
  title = {Chebyshev system},
  note = {Adler & Rivlin (1964), Proc. Amer. Math. Soc. 15, 794-796}
}