Pickover¶

Systems / Maps / Exotic maps

A Pickover (Clifford-style) 3-D ornamental attractor map.

discrete · map2 dimensionsattractor

Definition¶

@staticmethod
def _step(X, a, b, c, d):
    x, y = X
    xp = np.sin(a * y) + c * np.cos(a * x)
    yp = np.sin(b * x) + d * np.cos(b * y)
    return xp, yp

Parameters¶

Symbol	Default	Role
`a`	`-1.4`	angular frequency scaling the coordinates
`b`	`1.6`	angular frequency scaling the coordinates
`c`	`1`	amplitude of the cosine contribution
`d`	`0.7`	amplitude of the cosine contribution

State variables: x, y

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

1 fixed points
0 stable · 1 unstable

Use it in TSDynamics¶

import tsdynamics as ts

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

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

Reference¶

Pickover (1990), Computers, Pattern, Chaos and Beauty (St. Martin's Press)

BibTeX

@misc{pickover,
  title = {Pickover system},
  note = {Pickover (1990), Computers, Pattern, Chaos and Beauty (St. Martin's Press)}
}