dynamics of the fluctuations onto the invariant torus $\mathbb T^6$. The linearized dynamics is defined by $$\begin{aligned}
  \delta x(t) & \simeq & \sum_{\alpha=1}^3 (c_\alpha e^{\lambda_\alpha t} \mathring e_\alpha + d_\alpha e^{-\lambda_\alpha t} \mathring e_\alpha^*) \\
  \delta p(t) & \simeq & \sum_{\alpha=1}^3 (c_\alpha e^{\lambda_\alpha t} \hat e_\alpha + d_\alpha e^{-\lambda_\alpha t} \hat e_\alpha^*)\end{aligned}$$ where $\lambda_\alpha \in \imath \mathbb R$ are the Lyapunov values and $-I \hat e_\alpha = \lambda_\alpha^2 e_\alpha$, $\mathring e_\alpha = \lambda_\alpha \hat e_\alpha$ ($S e_\alpha = \lambda_\alpha e_\alpha$ and $S e_\alpha^* = - \lambda_\alpha e_\alpha^*$, the star denoting the complex conjugation). $$\begin{aligned}
  c_\alpha & = & \mathring e_\alpha^* \cdot \delta x(0) + \hat e_\alpha^* \cdot \delta p(0) \\
  d_\alpha & = & \mathring e_\alpha \cdot \overline{\delta x(0)} + \hat e_\alpha \cdot \overline{\delta p(0)}\end{aligned}$$ Let $\{\theta_\alpha\}_{\alpha \in \{1,...,6\}}$ be the local coordinates onto $\mathbb T^6$. Since the previous equations represent a phase trajectory wrapped around $\mathbb T^6$, the immersion equations of $\mathbb T^6$ into $\mathbb C^6$ are $$\begin{aligned}
  \delta x & \simeq & \sum_{\alpha=1}^3 (c_\alpha e^{\imath \theta_\alpha} \mathring e_\alpha + d_\alpha e^{\imath \theta_{\alpha+3}} \mathring e_\alpha^*) \\
  \delta p & \simeq & \sum_{\alpha=1}^3 (c_\alpha e^{\imath \theta_\alpha} \hat e_\alpha + d_\alpha e^{\imath \theta_{\alpha+3}} \hat e_\alpha^*)\end{aligned}$$

We rewrite these equations as $$\begin{aligned}
  