Equilibrium Bifurcations and Periodic Dynamics in a Planar Restricted Five-Body Problem with Four Unequal Primaries

We derive a general model for the planar restricted five-body problem with four unequal primaries in a non-symmetric coplanar central configuration. Four primaries form an admissible central configuration, and a fifth infinitesimal body moves in their gravitational field. We translate the primary configuration to its centre of mass and normalise the masses so that the primaries remain fixed in a frame rotating with unit angular velocity. We derive analytical conditions for the number and type of equilibrium points and for their spectral stability. We show that the number of nondegenerate equilibria is odd, that every generic fold creates or destroys one minimum and one saddle, and that spectral stability is possible only at a minimum. Numerical continuation on two separate positive-mass arcs gives as many as fifteen equilibrium points. Spectral stability occurs on a single equilibrium branch and is lost through a Hamiltonian–Hopf collision. We compute local Lyapunov families arising from the centre modes and classify them using their Floquet multipliers and Hill regions. We also obtain large-scale rosette-type circulating periodic families independently from resonant Kepler seeds.