Ph 621 Final Project

12/10/2003

The Henon Heiles Model

The Henon-Heiles Hamiltonian, developed to model the motion of stars in galaxies,

is:

where

Fixed Points

This model has four fixed points, which depend only on x and y.

This is demonstrated by showing that the gradient of the potential goes to zero at four points, and by determining what those points are.

The gradient of the potential is:

If the components of this are set to zero, the result is a homogenous system of equations in terms of x and y, one equation first order, one second order, whose solutions in the x – y plane are the four points:

(0,0),(0,1),,.

A plot of the equipotentials, with the four fixed points:

Stability of Fixed Points

The fixed point at the origin is stable: the other three are saddle points. This may be shown by looking at the determinant of the Jacobian of the map:

This determinant may be evaluated at the four fixed points. It is found that it is positive (=1) for the point at the origin, and negative (= -3) at the other three points.

The potential at or close to the three unstable fixed points is

V=,

where is zero or close to it.

Bounded and Unbounded Motion

From inspection of the contour plot, it is evident that contours for energies less than 1/6 near the origin are closed. It is also evident that the contours for potentials greater than 1/6, or for potentials less than 1/6 in the -120 or the 240 directions beyond the saddle points, are not closed. Hence we may conclude that motion for energies greater than 1/6 will not be bounded.

This behavior will perhaps be more evident from inspection of the plot below:

Poincare Sections

The equations of motion, from the Hamiltonian for this system, are:

These equations are coupled and nonlinear, and hence must be integrated numerically.

For energies < 1/6, the trajectories in the x, y, pY plane define a KAM torus:

The Poincare Section is the intersection of the trajectories or this KAM torus with the x=0 plane. In the plot above, the energy of the system is E = 0.06, and the initial x, y conditions are:

x= -0.1, y=-0.2, and pY =-0.05

The Poincare section for the above KAM torus looks like this:

Next is shown the KAM torus for initial conditions: x = 0, y = -.1475, pY = 0:

By trail and error, it was found that an initial condition of

would yield an initial very close to zero, for that energy. This is the outer contour in the next figure.

This contour is superimposed on the Poincare section for the KAM torus shown above.

At greater values of system energy, one begins to see the onset of chaotic behavior, at the hyperbolic fixed points of the system:

The hyperbolic fixed points are where the trajectories appear to cross. Such points are unstable fixed points, and it is there that one expects to see torii start to break up. The smearing at those points is an indication that the behavior is locally chaotic. In the spaces enclosed by the trajectories which cross, one would expect to see islands of stability, for the right initial conditions.

For somewhat larger energies one begins to see the KAM torii break up and form chains of islands of elliptic points linked by hyperbolic points:

At higher energies, the archipelagoes are well-defined:

In x/y/pY space, this “torus” looks like this:

A KAM torus like the above is probably quasiperiodic.

Above this energy, but still shy of E=1/6, the torii almost completely break down:

There may still be tiny elliptical islands near the fixed points.

Integrability

If the sign of the cubic term in the potential is changed, the character of the system behavior changes significantly.

The potential now has this shape:

The contour map looks like this:

The red points indicate potentials of 1/12, the blue 1/6, and the green 0.

A representative KAM torus of this map, for initial conditions:

looks like this:

The corresponding Poincare section is:

A torus for the same initial conditions, except now for energy = 1/13, is:

The Poincare section is:

For slightly greater energy (E=1/12), the torus now looks like this:

Again, for energy E = 1/10, the torus is:

It is evident from inspection of these torii that increasing the energy of the system does nothing to break them up. In other words, the system stays integrable.