Adding this exact control term to Hamiltonian (\[Ham1\]), the following invariant rotational torus is restored : $$\begin{aligned}
p(x,t)&=&\omega-\frac{\varepsilon_1 k_1}{\omega k_1-\omega_1} \cos
(k_1x-\omega_1t+\varphi_1) \nonumber
    \\[0.5\baselineskip]
&&-\frac{\varepsilon_2 k_2}{\omega
k_2-\omega_2}\cos(k_2x-\omega_2t+\varphi_2).\end{aligned}$$ This barrier of diffusion prevents a beam of particles to diffuse everywhere in phase space. We emphasize that the barrier persists for all the magnitudes of the waves $(\varepsilon_1,\varepsilon_2)$.

The control term (\[eqn:fTWT\]) has four Fourier modes, $(2k_1,-2\omega_1)$, $(2k_2,-2\omega_2)$, $((k_1+k_2),-(\omega_1+\omega_2))$ and $((k_1-k_2),-(\omega_1-\omega_2))$. If we want to restore an invariant torus in between the two primary resonances approximately located at $p\approx \omega_1/k_1$ and $p\approx
\omega_2/k_2$, the frequency $\omega$ has to be chosen between the two group velocities of the waves. If we consider a beam of particles with a velocity in between the velocities of the waves, i.e., $v_1=\omega_1/k_1$ and $v_2=\omega_2/k_2$, the main Fourier mode of the control term is $$\begin{aligned}
f_2&=&-\frac{\varepsilon_1\varepsilon_2 k_1 k_2}{2(\omega
k_1-\omega_1)(\omega k_2-\omega_2)} \nonumber
    \\[0.5\baselineskip]
&&\times \cos
[(k_1+k_2)x-(\omega_1+\omega_2)t+\varphi_1+\varphi_2].\end{aligned}$$

A convenient choice is $\omega=(v_1+v_2)/2$ and the control term is given by: $$f_2=\frac{2\varepsilon_1\varepsilon_2}{(v_1-v_2)^2}\cos
[(k_1+k_2)x-(\omega_1+\omega_2)t+\varphi_1+\varphi_2].\label{eqn:f2app}$$ Using this approximate control term does not guarantee the existence of an invariant torus. However, since the difference between $f$ given by Eq. (\[eqn:fTWT\]) and $f_2$ given by Eq. (\[eqn:f2app\]) is small, it is expected that for a Chirikov 