The conditions of field and voltage for inducing steady state excitations in fully perpendicular magnetic tunnel junctions (pMTJs), adapted for memory applications, were numerically investigated by the resolution of the Landau-Lifshitz-Gilbert equation in the macrospin approach. Both damping-like and the field-like spin transfer torque terms were taken into account in the simulations, as well as the contribution of the second order uniaxial anisotropy term (K$_2$), which has been recently revealed in MgO-based pMTJs. An in-plane applied magnetic field balances the out of plane symmetry of the pMTJ and allows the signal detection. Using this model, we assessed the states of the free layer magnetization as a function of strength of K$_2$ and polar θ$_H$ angle of the applied field (varied from 90° to 60°). There are two stable states, with the magnetization in-plane or out of plane of the layer, and two dynamic states with self-sustained oscillations, called in-plane precession state (IPP) or out of plane precession state (OPP). The IPP mode, with oscillation frequencies up to 7 GHz, appears only for positive voltages if θ$_H$ = 90°. However, it shows a more complex distribution when the field is slightly tilted out of plane. The OPP mode is excited only if K$_2$ is considered and reaches a maximum oscillation frequency of 15 GHz. Large areas of dynamic states with high frequencies are obtained for strong values of the field-like torque and K$_2$, when applying a slightly tilted external field toward the out of plane direction. The non-zero temperature does not modify the phase diagrams but reduces drastically the power spectral density peak amplitudes.