with the recreation of a set of invariant tori for $s=0.85$. However, this regularization apparently disappears when we consider the fully chaotic regime with $s=1.27$ (see Fig. \[fig:Poinc2\], panel (c)). Nevertheless the approximate control term has still a significant effect on the reduction of chaotic diffusion. This fact can be observed on the probability distribution functions of the electron beam velocity. This diagnostic will also be used in the experiment in order to see the effect of the control terms.

![(a): initial beam velocity distribution function for $s=0.85$. (b): final beam velocity distribution function without control term (\[eqn:fTWT\]) (light gray line), with the exact control term (black line) and with the approximate control term (\[eqn:fsimappTWT\])(dark gray line).[]{data-label="Distr_sp85"}](Fig4a.eps "fig:"){width="8.0cm" height="7.0cm"} ![(a): initial beam velocity distribution function for $s=0.85$. (b): final beam velocity distribution function without control term (\[eqn:fTWT\]) (light gray line), with the exact control term (black line) and with the approximate control term (\[eqn:fsimappTWT\])(dark gray line).[]{data-label="Distr_sp85"}](Fig4b.eps "fig:"){width="8.0cm" height="7.0cm"}

![(a): initial beam velocity distribution function for $s=1.27$. (b): final beam velocity distribution function without control term (\[eqn:fTWT\]) (light gray line), with the exact control term (black line) and with the approximate control term (\[eqn:fsimappTWT\]) (dark gray line).[]{data-label="Distr_s1p27"}](Fig5a.eps "fig:"){width="8.0cm" height="7.0cm"} ![(a): initial 