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, z)

projection (x, y, w)

Interactive: drag to rotate

Definition

\[ \begin{aligned} \dot{y_{0}} &= y_{2} \\ \dot{y_{1}} &= y_{3} \\ \dot{y_{2}} &= - 2 lam y_{0} y_{1} - y_{0} \\ \dot{y_{3}} &= - lam y_{0}^{2} + lam y_{1}^{2} - y_{1} \end{aligned} \]

Parameters

Symbol	Default	Role
lam	1	strength of the cubic anharmonic coupling

Properties

Lyapunov spectrum

$+0.01321,\; +0.01569,\; -0.01299,\; -0.01591$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 4$

Divergence ∇·f

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

Equilibria

1 equilibria
0 stable · 1 unstable

Define it in TSDynamics

import tsdynamics as ts

sys = ts.systems.HenonHeiles()
traj = sys.integrate(final_time=100.0, dt=0.01)

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

Reference

Hénon & Heiles (1964), Astron. J. 69, 73-79

BibTeX

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