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

\[ \begin{aligned} y_{0}' &= a - b y_{0}^{2} \end{aligned} \]

Parameters

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

Properties

Lyapunov spectrum

$+0.6922$
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

2 fixed points
0 stable · 2 unstable

Define it in TSDynamics

import tsdynamics as ts

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

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

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}
}