over the length of the TWT and the “devil’s staircase” is not yet evident [@escalier]. Two independent measures are shown by circles and squares to give an error estimate.[]{data-label="fig:const30b"}](Fig12.eps){width="7.5cm" height="7.2cm"}

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Experimental implementation of the control term {#experiment:implementation}
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We apply an oscillating signal at the frequency of 30 MHz on one antenna. It generates two waves: a helix mode with a phase velocity equal to $v_{\varphi}=4.07\times 10^6$ m/s, a beam mode with a phase velocity equal to the beam velocity $v_b$ (in fact two modes with pulsation $\omega=kv_b\pm\omega_b$ corresponding to the beam plasma mode with pulsation $\omega_b=(n_be^2/m\epsilon_0)^{1/2}$, Doppler shifted by the beam velocity $v_b$, merging in a single mode since $\omega_b\ll\omega$ in our conditions). Figures \[fig:085\] and \[fig:127\] (panel (b)) show the measured velocity distributions of the beam after interacting with these two modes over the length of the TWT for two different values of the Chirikov parameter. The case with $s=0.85$ was previously investigated  [@c18]. The red square (resp. blue triangle) shows the phase velocity $v_{\varphi}$ (resp.$v_{b}$) of the helix (resp. beam) mode on the middle of the resonant domain determined as the trapping velocity width of the helix mode $v_{\varphi}\pm 2\sqrt{eC_h\Phi/m}$ (resp. $v_b\pm
2\sqrt{eC_{b}\Phi/m}$ ) where $\Phi$ 