Henon¶

Systems / Maps / Chaotic maps

The Hénon map — a 2-D quadratic map whose folded attractor is the canonical strange attractor of dissipative maps.

discrete · map2 dimensionschaotic

Definition¶

\[ \begin{aligned} x' &= - a x^{2} + y + 1.0 \\ y' &= b x \end{aligned} \]

Parameters¶

Symbol	Default	Role
`a`	`1.4`	stretching
`b`	`0.3`	area contraction

State variables: x, y

Properties¶

Lyapunov spectrum

$+0.419,\; -1.623$
Sprott (2003), Chaos and Time-Series Analysis

Kaplan–Yorke dimension

$D_{KY} = 1.258$

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

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

Reference¶

Hénon (1976), Commun. Math. Phys. 50, 69-77
doi:10.1007/bf01608556

BibTeX

@misc{henon,
  title = {Henon system},
  note = {Hénon (1976), Commun. Math. Phys. 50, 69-77},
  doi = {10.1007/bf01608556}
}