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:

Figure 1.

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:

Figure 2.

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.



Figure 3.

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 4.

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 5.
 

Figure 6.
 

Figure 7.

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 8.




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 9.

Figure 10.

Figure 11.

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.

Figure 12.

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.