control term (\[eqn:fsimTWT\]) (panel (b)) and plus the approximate control term (\[eqn:fsimappTWT\]) (panel (c)).[]{data-label="fig:Poinc2"}](Fig3b.eps "fig:"){width="7.5cm" height="7.2cm"} ![Poincaré sections of Hamiltonian (\[Ham:sim\]) for $s=1.27$ without control term (panel (a)), plus the exact control term (\[eqn:fsimTWT\]) (panel (b)) and plus the approximate control term (\[eqn:fsimappTWT\]) (panel (c)).[]{data-label="fig:Poinc2"}](Fig3c.eps "fig:"){width="7.5cm" height="7.2cm"}

As expected, when the exact control term given by Eq. (\[eqn:fsimTWT\]) is added to the original Hamiltonian, then rotational invariant tori are restored. This is shown by the two Poincaré sections in Figs. \[fig:Poinc1\] and  \[fig:Poinc2\], panels (b), corresponding to the two values $s=0.85$ and $s=1.27$ of the Chirikov parameter. We notice that $s=1.27$ corresponds to a chaotic regime where the two resonances overlap according to Chirikov criterion ($s\ge 1$). Nevertheless the exact control term is able to reconstruct the invariant torus predicted by the method and to regularize a quite large region around the recreated invariant torus.

In order to study the effect of a simplified control term on the electron beam dynamics we perform numerical simulations adding the control term given by Eq. (\[eqn:fsimappTWT\]) to Hamiltonian (\[Ham:sim\]). As one can see from the Poincaré section depicted in Fig. \[fig:Poinc1\], panels (c), the effect of the approximate control term is still present 