-\frac{1}{2}\eta\ r$. This symmetry is, of course, present in the quantum problem also, where it will be exploited when calculating the Husimi distributions of the QEF’s. In the classical case, we exploit its existence to plot the stable manifolds around hyperbolic fixed points, which are otherwise very difficult to do because of their extreme sensitivity to perturbations.

Although the map is very complicated, there are a few periodic orbit cases that one can analytically study. By following the trajectory of the periodic orbit in phase space, and given the mapping equations, we can reconstruct the initial conditions giving rise to the orbit. For example, the fixed point shown in the Fig.5 (for $\ell=5$, ${\tilde\hbar}=0.008$, ${\tilde\omega_c}
=2\sqrt{2}$ and $\epsilon=1.0$), labeled F, is given by $r_0 = 0.75528003154206\ldots$, $p_0 = 2.43838534012017\ldots$.

Quantum to Classical correspondence {#subsubsec:sqfad}
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As mentioned in the introduction, one of the clear quantum manifestations of classical chaos (QMCC) emerges when one compares the spectral properties of specific model systems as appropriate parameters are tuned to classically produce a transition from integrable to completely chaotic regimes. In this section we follow the general thinking developed in Random Matrix Theories (RMT) to implement different tests to quantify the spectral properties 