Gauss¶

Systems / Maps / Polynomial maps

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

discrete · map1 dimensionchaotic

Gauss return map · bifurcation diagram — 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

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

1 fixed points
0 stable · 1 unstable

Use it in TSDynamics¶

import tsdynamics as ts

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

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

Reference¶

Hilborn (2000), Chaos and Nonlinear Dynamics, 2nd ed. (Oxford University Press)
doi:10.1093/acprof:oso/9780198507239.001.0001

BibTeX

@misc{gauss,
  title = {Gauss system},
  note = {Hilborn (2000), Chaos and Nonlinear Dynamics, 2nd ed. (Oxford University Press)},
  doi = {10.1093/acprof:oso/9780198507239.001.0001}
}