The final step when studying
a physical system is usually the resolution of a differential equation
system. In fluid mechanics, we face the Navier Stokes system.

Whereas on the one hand, there
are system that provides nice predictable solutions, on the other hand,
unfortunately, there are system that simply does not let them predict.
This is generally due to the presence of non-linear terms in those equation.
If you linearize them, you are of course more likely to solve the problem,
but you miss the infinite complexity of the physics.

The fact that a system seems to
be unpredictable is usually a matter of initial conditions sensitivity
(also called the butterfly effect). In other words, if you takes two points
whose characteristics are __almost__ the same, but not exactly, after
a while, the points will be so spaced out that you won't even think that
they have been close previously. This is what happen with a turbulent fluid
for instance.

This phenomena has long been thought
to be impossible to understand and the strange behaviors were thought to
be due to imprecision. Yet, in the middle of the 19's century, some scientists,
Feigenbaum, Lorentz or May, began to understand that this ** CHAOS**
was not a total disorder. This is in fact an organized chaos.

Even if they were then criticized by the scientific community, they keep on looking for the coherent structures in those systems.

From a more graphical point of view,
when drawing the portrait of phases of those systems, that is to say, the
final state of the system versus the evolution of the parameters that control
it, complex structures often appears.

Here is an example of such a graphics:
the evolution of a population of animals submitted to predator, diseases...

The picture represents the final
state (number of individual), versus the parameter that controls the constraint
(death causes):

- the logistic equation -

What is surprising is that even
if you zoom on a part of such a graphic, you find always the same degree
of complexity, whatever the power of your scaling factor.

Here is an illustration of that
:

- same complexity, whatever the scaling
factor -

To illustrate this degree of complexity,
we will study the ** Von Koch curve**, given that it is a very
good and very simple example of fractal drawing.

The word fractal comes from a mathematician, Benoît Mandelbrot. It refers to the Latin, 'frangere' ,which means 'to break'. It also reminds us of the non integer dimension of those forms.

The main topic of this document is then to calculate the Koch curve: