amplitude of the perturbation $V$ is large enough. The break-up of invariant tori leads to a loss of stability of the system until the last invariant torus of the integrable case is destroyed and then large scale diffusion occurs in phase space. In the case of a beam of charged particles whose dynamics is given by Hamiltonian (\[Ham1\]), for $\varepsilon_1$, $\varepsilon_2$ sufficiently large, an initially monokinetic beam will spread in velocity due to this diffusion.

Expression of the local control term {#loc:method}
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The aim is to provide an explicit expression for an additional perturbation such that a specific invariant torus is reconstructed in the modified system. We state here the main result which has been extensively described in Ref. [@chan06] : We consider Hamiltonian systems written as $$H({\bf A},{\bm\theta})={\bm \omega}\cdot {\bf A}+ W({\bf
A},{\bm\theta}),$$ where $\bm\omega$ is a non-resonant vector of ${\mathbb R}^d$. Without loss of generality, we consider a region near ${\bf A}={\bf 0}$ (by translation of the actions) and, since the Hamiltonian is nearly integrable, the perturbation $W$ has constant and linear parts in actions of order $\varepsilon$, i.e.$$\label{eqn:e4V} W({\bf A},{\bm\theta})=\varepsilon
v({\bm\theta})+\varepsilon {\bf w}({\bm\theta})\cdot {\bf
A}+Q({\bf A},{\bm\theta}),$$ where $Q$ is of order $O(\Vert {\bf A}\Vert ^2)$. We 