1 - INTRODUCTION: dynamic systems, chaos and fractals


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):

Logistic equation
- 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 :

Scaling factor
- 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:

Koch snowflake
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