chaotic regime ($s=1.27$). Concerning the approximate control term it is very efficient for $s=0.85$ while its efficiency is smaller in the strongly chaotic regime. However, it has still some regularizing effect, inducing the reconstruction of stable islands in phase space which can catch and thus confine a portion of the initial beam particles.

Robustness of the method {#loc:robust}
------------------------

The robustness of the control method for the case $s=1.27$ is studied with respect to an error on the phase or on the amplitude of the computed control term. In experiment, given the frequency $\omega_1+\omega_2$, the wave number $k_1+k_2$ of the control term does not satisfy in general the dispersion relation $k=K(\omega)$ since the dispersion relation is not linear. In our case it means that the experimentally implemented control term is not the exact one. For this reason we investigate the robustness of the control term given by Eq. (\[eqn:fsimappTWT\]) with a phase error $\varphi$, that is $$f(x,t)= 2\varepsilon^2r\cos[(k+1)x-kt+\varphi],
    \label{eqn:fappfasi}$$ and with an error on its amplitude ruled by a factor $\delta$, that is $$f(x,t)= 2\varepsilon^2r\delta\cos[(k+1)x-kt].
    \label{eqn:fappAmp}$$ The values given by Eq. (\[eqn:fsimappTWT\]) are $\varphi=0$ and $\delta=1$. In order to quantify the robustness of the approximate control term given by Eq. 