Temporal evolution, for the first 10 Kyr, of the canonical, osculating orbital elements, expressed in the Jacobian reference frame, for the PO configuration in Table 1. Top left panel: osculating period ratios for a subsequent pair of planets and their mean values (horizontal red lines), from top to bottom, ﹩{P}_{{\rm{b}}}/{P}_{{\rm{c}}}\simeq 2.170﹩, ﹩{P}_{{\rm{c}}}/{P}_{{\rm{d}}}\simeq 2.081﹩, and ﹩{P}_{{\rm{d}}}/{P}_{{\rm{e}}}\simeq 2.035﹩. We note that the proper orbital periods, in the sense of the mean motions as fundamental frequencies (Morbidelli 2002) expressed in Julian years of 365.25 days, are 52.36995, 104.73989, 209.47987, and 418.96324 for planets HR 8799e, d, c, and b, respectively, forming an exact 2:1, 2:1, 2:1 MMR chain. Top right panel: eccentricities of the planets HR 8799e, d, c, and b from top to bottom. Bottom left panel: one of the elements ﹩{x}_{i}\equiv {e}_{i}\cos {\varpi }_{i}﹩ (i = HR 8799b, c, d, and e) used to compute the secular frequency of the apsides rotation. The second component of the quasiperiodic signal (not shown) is ﹩{y}_{i}\equiv {e}_{i}\sin {\varpi }_{i}﹩. Bottom right panel: critical argument of the zeroth-order, four-body generalized Laplace resonance for the same initial condition that librates around ≃12°.