It may not seem very exciting but,
as we'll see it, there is a ton of "stuf" going on here. First of all,
no matter what initial point you plug in
for (x,y), virtually the same curve is traced. In fact,
the initial point must belongs to the attractor basin which is the set
of point whose all trajectories converge towards the attractor. Second,
the curve exhibits some rather interesting fractal characteristics. The
following graph shows the Henon's attractor : (go
to the program1)
It should be noted that the Henon's
attractor is not really a "curve" in the mathematical sense of the word.
It is actually a set of discrete points which appear to follow the pattern
of a curve (e.g the points are attracted to what looks like a smooth curve).
2-1 Closeups of the Henon Attracror
The blue rectangles on the previously figure indicate
the portion of the attractor that is zoomed-in on the frame following it
:
Those graphs shows a complex structur,
flaky which seems to be repeated indefinitely. So, we can reasonably think
that Henon's attractor is a
fractal structur and is a typical one-dimensional curve.
One might that its dimension would be slightly bigger than 1 and it is
just the case ! !
In fact, most textbooks and research papers cite the
dimension of the Henon attractor as around 1.26 .
2-2 Reduction of the areas
Another strange properties of the attractor
is that the area of the attractor is null. The followings graphs
can help to understand such a propertie.
Indeed, it shows how the area of the initial figure which
is a square decrease in function of the number of iterations.
After k iterations, the area will be reduce of
0.3^k. So, each once iterated areas will decrease with the ratio
of b=0.3. That results comes from the absolute value of the jacobian of
the transformation wich equal to b. We can deduce that the attractor
only re-cover a sub-set of the plan whose area is null. Here's the iteration
of a square by the Henon's function with a=1.4 and b=0.3 : (go
to the program1-1)
number of iteration = 0 number of iteration = 1 number of iteration = 2
number of iteration = 4 number of iteration = 8 number of iteration = 12
At
the twelfth iteration, we can already see a caracteristic figure to appear
: this is the Henon's attractor.
2-3 Correlations between x-values
The following graph show the correlation
between the samples x(i+io) and x(i) with io=[1,2,3,4,5,6]
. (go
to the program2)
The
dimension of the Henon's attractor in the space phase is two and the first
graph, which plot x(i) in function of x(i+1), allows to
reproduce the path of the attractor. This method is used
to look for attractors in dynamics systems if the dimension of the system
is known. For instance, if the dimension of the attractor was three,
we would have obtain the path of the attractor by ploting x(i) in
function of x(i+2). That's why the other graphs are different from the
path of the Henon's attractor.
2-4 Initial conditions
This is one of the most important properties
of strange attractors and show their chaotic behaviour. Two initial neighbooring
points will quickly drive appart and finally will not have the same behaviour
at all. For example, these two particles start at "almost" the same point
(0.5, 0.1 and 0.501, 0.099) but rapidly diverge over time . This shows
the sensitive dependence of Chaos on initial conditions. From a phisical
point of view, a little mistake on the measure will train very quickly
big errors on the calculus (like in meteorology). So, it is impossible
to anticapate the behaviour of a point in the long term.
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