\label{eqn:fsimTWT}$$ while the approximate control term given by Eq. (\[eqn:f2app\]) is $$f(x,t)= 2\varepsilon^2r\cos[(k+1)x-kt].
    \label{eqn:fsimappTWT}$$

![Poincaré sections of Hamiltonian (\[Ham:sim\]) for $s=0.85$ 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:Poinc1"}](Fig2a.eps "fig:"){width="7.5cm" height="7.2cm"} ![Poincaré sections of Hamiltonian (\[Ham:sim\]) for $s=0.85$ 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:Poinc1"}](Fig2b.eps "fig:"){width="7.5cm" height="7.2cm"} ![Poincaré sections of Hamiltonian (\[Ham:sim\]) for $s=0.85$ 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:Poinc1"}](Fig2c.eps "fig:"){width="7.5cm" height="7.2cm"}

Poincaré sections of Hamiltonian (\[Ham:sim\]) computed for two values of the Chirikov parameter, $s=0.85$ and $s=1.27$, are depicted in Figs. \[fig:Poinc1\]-\[fig:Poinc2\], panels (a). We notice that in both cases no rotational invariant tori survive and therefore trajectories can diffuse over the whole phase space in between the two primary resonances.

![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"}](Fig3a.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 