Gauss

Systems / Maps / Polynomial maps

The Gauss (continued-fraction) map — number-theoretic chaos on the unit interval.

discrete · map1 dimensionchaotic

return map · bifurcation diagram

Definition

@staticmethod
def _step(X, a, b):
    x = X
    return np.exp(-a * x**2) + b

Parameters

Symbol	Default	Role
a	4.9	width of the Gaussian (larger a = narrower peak)
b	-0.5	vertical shift controlling the bifurcation cascade

Properties

Lyapunov spectrum

$+0.1746$
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

1 fixed points
0 stable · 1 unstable

Define it in TSDynamics

import tsdynamics as ts

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

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

Reference

Hilborn (2000), Chaos and Nonlinear Dynamics, 2nd ed. (Oxford University Press)

BibTeX

@misc{gauss,
  title = {Gauss system},
  note = {Hilborn (2000), Chaos and Nonlinear Dynamics, 2nd ed. (Oxford University Press)}
}