Logistic

Systems / Maps / Population maps

The logistic map — one quadratic line whose period-doubling cascade is the textbook road to chaos.

discrete · map1 dimensionperiod-doublingchaotic

return map · bifurcation diagram

Definition

\[ \begin{aligned} x' &= r x \left(1 - x\right) \end{aligned} \]

Parameters

Symbol	Default	Role
r	3.9	growth rate (the bifurcation parameter)

State variables: x

Properties

Lyapunov spectrum

$+0.6931$
exact result at r = 4

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.Logistic()
traj = sys.iterate(steps=10_000)

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

Reference

May (1976), Nature 261, 459-467

BibTeX

@misc{logistic,
  title = {Logistic system},
  note = {May (1976), Nature 261, 459-467}
}