Numerical results

1/ The attractor and it's fractal structure :

As said in the presentation of the model, this is the result of 10 000 iterations with the fortran code provided. Initial conditions are (X,Y)=(0,0) and the parameters a and b were fixed to their standard value : a=1.4 and b=0.3. This figure is called an attractor, because any initial conditions in the bassin of attraction converge to the attractor.

The Henon attractor has a fractal structure, which makes it a "strange attractor". It means that if you zoom on a region of the attractor, you will see the same shapes that the ones on the larger view. To understand a bit more what this means, let's have a look at the part of the attractor inside the box :

We can see three groups of lines, a group of three lines, a group of two and a single line. Let's now zoom on the group of three lines in the box :

The structure is exactly the same that the one on the previous figure!! What about zooming again on the group of three lines?

Once again, there are three same groups of lines. This property of keeping the same shape at any scale is typical of fractal objects.

2/ Sensibility to initial conditions :

An other characteristic of cahotic systems is an exponential sensibility to initial conditions (SIC). It means that if you start to iterate with two points that are distant of e (chosen as small as you want), the distance between their images wil increase enponentially until it reaches the size of the attractor.

To demonsrate this property in the case of Henon attractor, two simulations were ran with initial conditions distant of 1e-15. You can see below the distance between the points ploted every iteration :

This numerical experiment gives a the aproximate relation between N (number of iterations) and D (distance between the points) : D=exp(6N).

3/ Deformation of a circle :

To briefly illustrate the influence of the contraction rate b, we shall have a look at what happens to a circle when iterated in Henon model :

4/ Bassin of attraction :

The fortran code can draw the bassin of attraction of the model with any values of the parameters.
We noticed that if a point is not in the bassin of attraction, it quickly diverges (over 1e300 in less than 15 iterations !). Consequently, the method to draw the bassin is to test a grid of points and plot them only if their image has not diverged in less than a hundred iterations. To have an idea of what the result should be (the next image was found on the internet), here is the bassin (in black) and the attractor (in white) :

And now have a look at what came out of the code:

To have a larger view, a wider range of coodinates was explored:

5/ Others funny attractors with Henon model :

In all of the above, a and b are always set to their "standard values: a=1.5 and b=0.3. Therefore, we are now going to have a quick look at what happens when these values are changed.

Initial conditions (x,y)=(0,0), a=-0.52, b=-0.998.

Initial conditions (x,y)=(0,0), a=-0.52, b=-1. Please notice that this experiment is very similar to the previous one, only b differs of 0.002!! And yet , the shapes are very different.

Initial conditions (x,y)=(0.132,-1169), a=-0.651, b=-1.

Initial conditions (x,y)=(-0.313,-0.166), a=-0.497, b=-0.999.

Initial conditions (x,y)=(0.22,0.01), a=0.999, b=-0.999.

6/ Test of the fixed points :

The best way to test the fixed points might be to have a try with the fortran code, however, you may find here the values of the coordinates for the first 100 iterations.
A funny thing happens: one of the two points leaves it's original position after 25 iterations whereas the other remains... who's fault???

Note that the code also calculates the eigen values at the fixed points to know wether the points are stable or not.

Now, you may wonder how these graphs were processed... so please have a look at the fortran code that was written. You may also want to go back to the main page.