Dynamical regimes of two eccentric and mutually inclined giant planets

We consider a basic planetary system composed by a Sun like star, a Jupiter-like planet and a Neptune-like planet in a wide range of orbital configurations not limited to the hierarchical case. We present atlases of resonances showing the domains of $\sim$ $1300$ mutual mean-motion resonances (MMRs) and their link to chaotic and regular dynamics. Following a semi-analytical method for the study of the secular dynamics we found two regimes for equilibrium configurations: one for low mutual inclinations were equilibrium is related to oscillations of $\Delta \varpi$ around $0°$ or $180°$, and other for high mutual inclinations where the equilibrium is given by defined values of the ${\omega }_i$ equal to integer multiples of $90°$. By numerical integration of the full equations of motion we calculate the fundamental frequencies of the systems in their diverse configurations and study their dependence with the orbital elements. According to the analysis of the fundamental frequencies we found two dynamical regimes depending on the initial mutual inclination and the limit between the two regimes occurs at some critical inclination $30°\lesssim i_c\lesssim 40°$ defined by the occurrence of the secular resonance $g_1=g_2$. For $i<i_c$ the dynamics is analogue to the classic secular model for low $(e,i)$ with well defined three fundamental frequencies and free and forced modes, conserving quasi constant the mutual inclination. For $i>i_c$ the dynamics is completely different with increasing changes in mutual inclination and emerging combinations of the fundamental frequencies and, depending on the case, dominated by the secular resonance or the vZLK mechanism.