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As a conclusion, I would like to recapitulate all the bifurcations I have meet in this project.
The figure 3.6 allows us to have a more syntetic vue of the bifurcations of the KSE.

Figure 3.6: Bifurcation diagram
\includegraphics [scale=0.8]{bd.eps}

In fact, I have seen that, even if for low value of $ \nu $, the theoretical study can predict the dynamic of the Kuramoto-Sivashinsky, the more $ \nu $ increase, the more complex the dynamic is. The dynamic of the KSE is a big problem because this equation is representative of lot of phenomenon like :

That's why this problem is very studied.

Moreover, for the course of hydrodynamic instabilities, this equation was very interesting. This project allows me to see the applications of this course and make me work on a important part of my formation : the numerical methods.

Julien Delbove