Ulam

Systems / Maps / Chaotic maps

The Ulam map (logistic at r=4) — exactly chaotic with a known invariant density.

discrete · map1 dimensionchaotic

return map · bifurcation diagram

Definition

\[ \begin{aligned} x' &= a - b x^{2} \end{aligned} \]

Parameters

Symbol	Default	Role
a	1	map offset (a=1 gives the ergodic regime)
b	2	quadratic gain (b=2 gives the ergodic regime)

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

2 fixed points
0 stable · 2 unstable

Use it in TSDynamics

import tsdynamics as ts

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

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

Reference

Ulam & von Neumann (1947), Bull. Amer. Math. Soc. 53, 1120

BibTeX

@misc{ulam,
  title = {Ulam system},
  note = {Ulam & von Neumann (1947), Bull. Amer. Math. Soc. 53, 1120}
}