## 4 The Phase Space Structure of the Main Terms
### Single Resonance Hypothesis Description  
If we apply the above description to the Hamiltonian Equation (7), the actions are \(\mathbf{I}=(L,G,H)\), the angles \(\mathbf{\psi}=(\ell,g,h)\) and the unperturbed term \(\mathcal{H}_{0}\) is given by the \(J_{2}\)-term and both the lunar and solar mean terms, namely  
\[\mathcal{H}_{0}=\mathcal{H}_{J_{2}}(G,H;L)+C_{0}^{\mathbb{Q}}\ \mathcal{A}_{0}^{ \mathbb{Q}}\ (G,H;L)+C_{0}^{\odot}\mathcal{A}_{0}^{\odot}(G,H;L). \tag{27}\]  
The resonant perturbation is given by the Hamiltonian contribution of a lunisolar dominant harmonic related to the resonant angle, considered on a case-by-case basis. This is to say that, in order to start from a simplified analysis, we assume that only one lunisolar term affects the dynamics at a time. Under the hypothesis of isolated resonance, the dynamics in a small enough neighborhood are described by a resonant Hamiltonian of the form given in Equation (17). For each case that we will show, after performing a coordinate change of the form shown in Equation (21), the motion evolves in the \((J_{1},\theta_{1})\) plane. The first integral of motion is Kozai-like, i.e., the evolution of \(e\) and \(i\) is coupled (recalling that the semi-major axis is constant in our assumptions). Thus, the phase portraits are computed assuming \(L=L_{moln}\) and they correspond to the given integral of motion computed at the Molniya parameters. The resonant dominant harmonics for which the SRM gives a reliable description of the phase plane structure are listed in Table 8, while the resonances associated with the arguments in Table 9 exhibit a non-standard behavior. In Table 8, we show the center of libration related to each resonant harmonic and the corresponding maximum width, as computed with the standard approach (SRM) through Equation (23). The maximum real excursion in eccentricity and in inclination, computed by using Equation (26), gives substantially the same width obtained with the SRM. These facts can be appreciated by looking at the phase portraits from Figures 1-4. The dynamical structure arising from the pendulum-like Hamiltonian is depicted on the left, while, on the right, we show the results obtained from the resonant Hamiltonian not developed in Taylor series; the Y-axis is always converted in eccentricity or in inclination. \begin{table}
\begin{tabular}{l l l l l l l l l} \hline \hline
**Critical Argument** & **First Integral** & \(\mathbf{e^{*}}\) & \(\mathbf{e_{s}}\) & \(\mathbf{i^{*}}\) & \(\mathbf{i_{s}}\) & \(|I_{\mathbf{Is}}-I_{\mathbf{In}}|\) & \(\mathbf{\Delta e}\) & \(\mathbf{\Delta i}\) \\ \hline \(2g-h\) & \(\sqrt{a(1-e^{2})}(\cos i+\frac{1}{2})\) & 0.64 & 0.64 & 69.14 & 69.03 & 114.41 & 0.13 & 7.3 \\ \(2g+h\) & \(\sqrt{a(1-e^{2})}(\frac{1}{2}-\cos i)\) & 0.98 & 0.98 & 56.06 & 56.06 & 1.20 & 0.004 & 0.55 \\ \(2g+h-h_{\mathbb{Q}}\) & \(\Gamma-\frac{1}{2}\sqrt{a(1-e^{2})}\) & & & & & & & \\ \(\sqrt{a(1-e^{2})}\cos i\) & & & & & & & & \\ \(\sqrt{a(1-e^{2})}\cos i\) & & & & & & & & \\ \(\sqrt{a(1-e^{2})}\cos i\) & & & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Resonances whose dynamics are well described by the SRM. The first column identifies the resonances through the critical argument associated with it; the second column shows the first integral arising from the resonant Hamiltonian. \(\Gamma\) is the dummy momentum introduced to add one dimension to the phase space in case of resonances involving the lunar ascending node. The center of libration is given in terms of Keplerian elements: \((e^{s},i^{s})\) identifies the libration center of the exact resonance while \((e_{s},i_{s})\) gives the center of libration related to the real resonance. \(|J_{\mathrm{Is}}-J_{\mathbf{In}}|\) (in km\({}^{2}\)/s) gives information about the asymmetry of the resonant region. The last two columns show the libration width in terms of \(e\) and \(i\); the same values are obtained with the SRM Equation (23) and with NSA Equation (26). The values in inclination are reported in degrees.