III.   Exploitation of the results.

Now that the program works, it is possible to study the henon's attractor. The most common form of the Henon's attractor has a banana shape. It appears for the parameters a = 1.4 and b = 0.3. The image given by the program is :
 

This shape is not continuous but it is composed of a lot of points. The program loops for a lot of initial conditions and the shape is still the same. All the values converge with different manners but the result is always the same. Lot a small structures appear while zooming :

Even if this zoom is not very nice, we can see different attractive orbits appearing. In fact, each orbit deals with other orbits but the resolution is not good enough to see them. On closer inspection, the shape is made up of thicker and thinner parts.

By testing the program with other parameters values, other strange attractors appear. For example, with the parameters a = -0.52 and b = -0.998, the result obtained is totally different :

And a little modification in these parameters change the behaviour of the whole attractor. For a = -0.52 (same as before) and b = -0.9999, the result is surprising :

That is what is called a bifurcation, a small change in the parameters implies total change of behaviour for the system.
 
 

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