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

.