Chebyshev¶

Systems / Maps / Geometric maps

Iterated Chebyshev polynomials — exactly chaotic maps of the interval.

discrete · map1 dimensionchaotic

Chebyshev return map · bifurcation diagram — 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)

State variables: x

Properties¶

Lyapunov spectrum

TODO — lyapunov_spectrum failed: AttributeError

Kaplan–Yorke dimension

TODO — requires a numeric Lyapunov spectrum

Divergence ∇·f

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

Equilibria

5 fixed points
0 stable · 5 unstable

Use it in TSDynamics¶

import tsdynamics as ts

system = ts.systems.Chebyshev()
traj = system.run(steps=10_000)
ts.plot(traj)

spectrum = ts.analysis.lyapunov_spectrum(system)
ts.analysis.kaplan_yorke_dimension(spectrum)

Reference¶

Adler & Rivlin (1964), Proc. Amer. Math. Soc. 15, 794-796
doi:10.1090/s0002-9939-1964-0202968-3

BibTeX

@misc{chebyshev,
  title = {Chebyshev system},
  note = {Adler & Rivlin (1964), Proc. Amer. Math. Soc. 15, 794-796},
  doi = {10.1090/s0002-9939-1964-0202968-3}
}