HenonHeiles

Systems / ODEs / Chaotic attractors

A galactic potential whose bounded motion transitions from regular tori to chaos with energy.

continuous · ODE4 dimensionsconservativechaotic

projection (x, y, px)

projection (x, y, py)

Interactive: drag to rotate

Definition

\[ \begin{aligned} \dot{x} &= px \\ \dot{y} &= py \\ \dot{px} &= - 2 lam x y - x \\ \dot{py} &= - lam x^{2} + lam y^{2} - y \end{aligned} \]

Parameters

Symbol	Default	Role
lam	1	strength of the cubic anharmonic coupling

State variables: x, y, px, py

Properties

Lyapunov spectrum

TODO — lyapunov_spectrum failed: AttributeError

Kaplan–Yorke dimension

TODO — requires a numeric Lyapunov spectrum

Divergence ∇·f

$\nabla\!\cdot f = 0$
constant

Equilibria

3 equilibria
0 stable · 3 unstable

Use it in TSDynamics

import tsdynamics as ts

system = ts.systems.HenonHeiles()
traj = system.run(final_time=100.0, dt=0.01)
ts.plot(traj)

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

Reference

Hénon & Heiles (1964), Astron. J. 69, 73-79
doi:10.1086/109234

BibTeX

@misc{henonheiles,
  title = {HenonHeiles system},
  note = {Hénon & Heiles (1964), Astron. J. 69, 73-79},
  doi = {10.1086/109234}
}