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Instability of the second order modes

The theoretical part of this report shows that the second order modes become unstable for $ \nu $ greater than $ (4 \pi)^2 $. Nevertheless, saying that a mode is unstable is not equivalent to saying that the calculation will converge. For those values of $ \nu $ we have a example of this behaviour. In fact a second order perturbation is effectively amplify but I can't say if the solution will converge or not. It's a particularity fo this zone. Sometimes the solution solution will converge and sometimes we have an example of chaotic system.

The figure 3.5 shows a converge solution.

Figure 3.5: Solution for $ \nu = 200$
\includegraphics [scale=0.8]{sol200.eps}

The exact determination of the zone of convergence would be very hard, and has not been done in this project.

Julien Delbove