## 4 The Phase Space Structure of the Main Terms
### Single Resonance Hypothesis Description
We note that the phase portraits for the cases where the angle does not include the lunar node have been computed considering the contribution of both Sun and Moon. Moreover, since the harmonic argument depending on the lunar ascending node leads to a non-autonomous resonant Hamiltonian, it is necessary to introduce a dummy momentum \(\Gamma\) and a new conjugate angle depending on the lunar node in order to eliminate the explicit linear time dependency. For this reason, there are two first integrals in correspondence of \(2g\pm h_{\mathbb{G}}\), \(2g+h-h_{\mathbb{G}}\) in Tables 8 and 9. Figure 1: Phase space associated with the \(2\hat{g}\) dynamics. Contour plot of the pendulum-like approximation (plot title SRM on the left) and of the resonant Hamiltonian not developed in Taylor series (plot title NSA on the right). The X-axis always shows the critical angle \(2g\) in degrees. The Y-axis is converted in inclination (on the top), and measured in degrees, or in eccentricity (on the bottom). _Green_ lines denote the librating curves around the stable equilibrium, _purple_ lines denote the circulation region, while the separatrices are depicted in _red_. \begin{table}
\begin{tabular}{l l l l l l l l l} \hline
**Critical Argument** & **First Integral** & \(\mathbf{\varepsilon_{s}}\) & \(\mathbf{\varepsilon_{u}}\) & \(\mathbf{i_{s}}\) & \(i_{u}\) & \(|\mathbf{J_{ls}}-\mathbf{J_{ls}}|\) & \([\mathbf{\varepsilon_{min}},\mathbf{\varepsilon_{max}}]\) & \([\mathbf{\varepsilon_{min}},\mathbf{\varepsilon_{max}}]\) \\ \hline \(2g\) & \(\sqrt{a(1-e^{2})}\cos i\) & 0.72 & 0.71 & 63.29 & 63.69 & 625.10 & \([0.55,0.79]\) & \([59.30,68.11]\) \\  & \(\sqrt{a(1-e^{2})}\cos i\) & 0.62 & 0.60 & 67.97 & 68.48 & & & \\  & \(\Gamma+\frac{1}{2}\sqrt{a(1-e^{2})}\) & 0.57 & 0.54 & 68.95 & 69.57 & & & \\ \hline \end{tabular}
\end{table} 
Table 9: Resonances whose dynamics are not appropriately described by the SRM. The first column identifies the resonance through the associated argument; the second column shows the first integral arising from the resonant Hamiltonian. The equilibria are given in terms of eccentricity and inclination: \((\varepsilon_{s},i_{s})\) stable ones, \((\varepsilon_{u},i_{u})\) unstable ones. \(|J_{ls}-I_{ls}|\) (in km\({}^{2}\)/s) gives information about the asymmetry of the resonant region. The last two columns show the maximum excursion, in terms of \(e\) and \(i\), that may be attained in the libration region as computed with NSA through Equation (26). The values in inclination are reported in degrees. N.E. means “not evaluated”. Figure 3: _Cont._  
Figure 2: Phase space associated with the \(2\delta+h\) dynamics. Contour plot of the pendulum-like approximation (SRM on the left) and of the resonant Hamiltonian not developed in Taylor series (NSA on the right). The X-axis always shows the critical angle \(2g+h\) in degrees. The Y-axis is converted in \(i\) (on the top), and measured in degrees, or in \(e\) (on the bottom). _Green_ lines denote the librating curves around the stable equilibrium, _purple_ lines denote the circulation region, while the separatrices are depicted in _red_. The lunisolar periodic component with argument \(2g\) produces a non-negligible contribution to the dynamics of the argument of the pericenter, if compared with the oblateness effect and with the precession due to the lunisolar mean terms. Therefore, the ideal model SRM does not give a reliable estimate of the resonant region, in accordance with [5]. Figure 1 depicts the dynamics in the \((e,2g)\) plane and in the \((i,2g)\) plane around the main resonance. Figure 4: Phase space associated with the \(2g-h\) and \(2g+h-h_{\mathcal{Q}}\) (right) dynamics. The X-axis always shows the critical angle in degrees. The Y-axis is converted in \(i\) (on the top), and measured in degrees, or in \(e\) (on the bottom). _Green_ lines denote the librating curves around the stable equilibrium, _purple_ lines denote the circulation region, while the separatrices are depicted in _red_.