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

Use it in TSDynamics

import tsdynamics as ts

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

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

Reference

May (1976), Nature 261, 459-467
doi:10.1038/261459a0

BibTeX

@misc{logistic,
  title = {Logistic system},
  note = {May (1976), Nature 261, 459-467},
  doi = {10.1038/261459a0}
}