EXAMPLES OF THE HENON ATTRACTOR DYNAMICS

Evolution of an infinitesimal distance between 2 initial points

The calculation of the distance delta introduced in the initial values of the Henon transformation is based on the calculation of 2 series which initial points are closed to the attractor and separated by an initial infinitesimal distance delta0: we chosed delta0 = 10^-8. The main value of the distance delta at the iteration number i growths like
delta = delta0*exp(lambda1*i), where lambda1 is the highest of the Lyapunov exponents. An evaluation of lambda1 is given by the rate of the semi-log scale graph delta versus k.
To get this curve, we used the following Matlab program:

This program can be found in /users/mfn-u/mfn07/instab_hydro/distance.m

Here is the semi-log scale graph delta versus k:

Evolution of an infinitesimal distance between 2 initial points.

We find that the mean value of  lambda1 is lambda1=0.5, which with the theory. After 40 iterations, the distance between 2 following points doesn't grow anymore: we reach the scale of the attractor.

Deformation of a circle

Another example of the dynamics of this strange attractor is the deformation of a circle, it is also an illustration of area contraction.. We take an initial serie of points regularly placed on a circle close to the attractor, and we iterate 2 times. Here is the Matlab program we used:

This program can be found in /users/mfn-u/mfn07/instab_hydro/circle.m

On the following picture, we can see: in blue, the initial circle (iteration 0), its centre is not on the attractor, in green the ensemble of points we get after the first iteration, and in cyan the ensemble of points we get after the second iteration, in red we have the shape of the attractor. We observe the area contraction, the ensemble of points is attracted and gets like stretched.

Deformation of a circle when the center of the initial circle is not on the attractor.

If we take now the center of the initial circle on the attractor, after an interation, we got the green ensemble, an ellipse, and after another iteration we get the cyan ensemble of points. We notice the stretching effect along the attractor, and the contracting effect perpendicularly to it.

Deformation of a circle when the center of the initial circle is on the attractor.