this experimental apparatus can be accurately represented using a Hamiltonian which describes the motion of a charged particle (with unit mass) interacting with two electrostatic waves [@c16] :

$$\begin{aligned}
\label{eqn:H}
    H(p,x,t)&=&\frac{p^2}{2}+\varepsilon_1\cos (k_1 x-\omega_1
    t+\varphi_1)\nonumber
    \\[0.5\baselineskip]
    &&+\varepsilon_2\cos (k_2 x-\omega_2
    t+\varphi_2),
    \label{Ham1}\end{aligned}$$

where $(p,x)\in {\mathbb R}\times [0,L]$ are the momentum and position of the particle in a tube of length $L$. The amplitudes, wave numbers, frequencies and phases of the two waves are denoted respectively $\varepsilon_i$, $k_i$, $\omega_i$ and $\varphi_i$ for $i=1,2$. We notice that the beam intensity is sufficiently low such that the wave growth rate is negligible upon the length of the experiment that is we are in the test-particle regime.

Generically, the dynamics of the particles governed by Hamiltonian (\[Ham1\]) is a mixture of regular and chaotic behaviors, mainly depending on the amplitudes of the waves. The Chirikov parameter [@Chirikov] defined as the ratio between the two half-width of the primary resonances by the distance between these resonances, i.e., $$\label{eqn:pCh}
    s=\frac{{2}(\sqrt{{|\varepsilon_1}|}+\sqrt{{|\varepsilon_2|}})}{|{\omega_2}/{k_2}-{\omega_1}/{k_1}|},$$ gives a first rough approximation of the chaoticity degree of the system. Hamiltonian (\[eqn:H\]) has a typical behavior of integrable system for small values of this parameter ($s$$\ll1$). For large enough amplitudes of the waves ($s\sim 1$), large 