Fractal structure of the Henon map
The Henon map is governed by the following iterative system of real numbers (xn ,yn ):
xn+1 = 1 -1.4 xn2 + yn
yn+1 = 0.3 xn
For initial values located in the basin of attraction, the points converge very quickly to a particular structure, which can be qualified of a strange attractor.
Matlab program to draw the Henon attractor
We used a Matlab program to calculate and draw this structure. The program is shown below:
This program can be found and run in /users/mfn-u/mfn07/instab_hydro/henon1.m
Permanent structure of the attractor
We used 100000 points to draw the Henon attractor, with the initial conditions x(1)=y(1)=0:
The points do not flow continuously, but hop from one location to another: the way they do is irregular and chaotic. After a few iterations, 2 points which were initially very close lead to an exponential growth in term of distance ( see Evolution of an infinitesimal distance between 2 initial points ).
The Henon attractor also shows a great deal of fine structures (an infinite amount to be exact). Any cross-section through an arm of the Henon attractor is equivalent to a Cantor middle thirds set. Successive magnifications show an ever increasing degree of detail.
We can see 3 parallel lines, the upper line seems to be thicker:
If we do a second magnification we can see ( Figure 3 ) that the upper line observed in the first magnification seems to be made of 3 lines, with the upper one thicker. The middle line seems to be made of 2 lines, and the bottom one seems to be made of 1 line.
If we make a magnification of the 3 upper lines in Figure 3, again we notice the same phenomenon: the upper line seems to be made of 3 lines, the middle line seems to be made of 2 lines, and the bottom line seems to be made of 1 line.
Figure 5, 6 and 7 clearly show that the groups of lines in Figure 4 are the finest structures in the attractor. But our process is limited by the finite number of points in our simulation: if we could have an infinite number of points we would have an infinite number of sub-structures in the attractor. It is the fractal dimension.
Figure 8 shows a closure of both middle and bottom group of lines in Figure 3: we clearly see that the middle group of lines is made of 2 lines, and the bottom group of lines is made with 2 lines very close to each other.
Figure 9 is a closure of the right edge of the entire attractor ( see Figure 1): we can see how the middle main lines vanishes: in fact this line is closed. As shown on Figures 10 and 11, this closed line is made of sub-structures of finer lines. Again, the number of sub-structures is limited by the finite number of points we took for our simulation.
Figure 12 is a closure of the bottom edge of the entire attractor ( see Figure 1): again we can see that the line is closed and made of a few sub-structures of lines.
These experiments illustrate the fact that the structure of this attractor is repeating identically at each observation scale. This permanent structure at different scales is caracteristic of a fractal object.