WOLFRAM|DEMONSTRATIONS PROJECT

The Hénon-Heiles System

​
t
400
x
0
-0.1
y
0
-0.2
p
x
0.257229
p
y
-0.05
energy = 0.06
​In 1964, Henon and Heiles introduced a model to study the motion of a star through a galaxy with a two-dimensional, non-integrable potential, the result of adding a perturbation to an integrable harmonic potential. This Demonstration shows a plot of the phase-space trajectory and the Poincaré map. This system is an often-cited example of chaotic dynamics.
The equations of motion are

x
=
p
x
,

y
=
p
y
,

p
x
=-x-2xy
, and

p
y
=-y-
2
x
+
2
y
, where
x
and
y
are the coordinates and
p
x
and
p
y
are the momenta. The energy is equal to
2
p
x
2+
2
p
y
2+
2
x
/2+
2
x
y+
2
y
2-
3
y
3
.