long term stability suggests that the dynamical system described by $(|\delta x_i\rangle)_i$ presents an invariant manifold.

Existence of an invariant torus {#invtorus}
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We consider the nonlinear equation \[equfluct\] where we treat $\{X_i\}_i$ and $\{x_i\}_i$ as being constant (the variations of these quantities are supposed very slow compared to the variations of $\{|\delta x_i \rangle\}_i$ and we consider them as being frozen in a first time).\
We restrict our attention to the case $N=1$ which is instructive for our discussion, $|\delta x_i \rangle$ is then reduced to be a complex scalar $\delta x_i \in \mathbb C$ obeing to the nonlinear equation: $$\label{eqsd}
  \delta \ddot x_i = -\|X-x\|^2 \delta x_i + (X^j-x^j)(X_i-x_i)\delta x_j - 2 \overline{\delta x^j} \delta x_j \delta x_i + 2 \overline{\delta x_i} \delta x^j \delta x_j$$ We can rewritte this equation as $$\left\{ \begin{array}{rcl} \delta \dot x_i & = & F_i(\delta p) = \delta p_i \\ \delta \dot p_i & = & F_{\hat i}(\delta x) \end{array} \right.$$ where $\delta p_i \in \mathbb C$ is the momentum associated with $\delta x_i$, $F_{\hat i}(\delta x)$ being the right hand side of the equation \[eqsd\]. We adopt the notations such that the indices $i$, $\hat i$, $\bar i$ and $\tilde i$ 