Hopalong¶

Systems / Maps / Exotic maps

Barry Martin's "Hopalong" map — a square-root-and-sign nonlinearity that sprays iterates into layered ornamental chaotic attractors.

discrete · map2 dimensionsattractor

Definition¶

@staticmethod
def _step(X, a, b, c):
    x, y = X
    xp = y - 1 - np.sqrt(np.abs(b * x - 1 - c)) * np.sign(x - 1)
    yp = a - x - 1
    return xp, yp

Parameters¶

Symbol	Default	Role
`a`	`3.1`	additive constant in the y-recurrence (overall offset)
`b`	`2.5`	scale inside the square-root argument
`c`	`4.2`	shift inside the square-root argument

Properties¶

Lyapunov spectrum

$+0.04241,\; -0.04241$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 2$

Divergence ∇·f

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

Equilibria

none found (no fixed points)

Define it in TSDynamics¶

import tsdynamics as ts

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

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

Reference¶

Dewdney (1986), Scientific American 255(3), 14-20

BibTeX

@misc{hopalong,
  title = {Hopalong system},
  note = {Dewdney (1986), Scientific American 255(3), 14-20}
}