No. Celestial mechanics (Newtonian or GR) relies on a static or slowly varying central inverse-square potential:
Orbits arise from the balance of attraction and centrifugal terms in an effective 1-D potential, producing (nearly) closed ellipses. There is no requirement for rapid parametric oscillation of a quadrupole saddle.
A 2025 arXiv paper explores a special-case non-orbital trapping of charged particles near the center of equal-mass charged binary black holes by treating the near-zone potential as a rotating saddle. This is an analogy for dynamic stability under fine-tuned conditions, not a description of ordinary planetary or stellar orbital gravity.