Nonlinear Stability, Resonances, and Singular Reduction in the Unequal-Mass Equilateral Restricted Four-Body Problem

Abstract

We study the nonlinear stability of the elliptic equilibrium points $L_3$, $L_5$, and $L_6$ in the planar equilateral restricted four-body problem with unequal primary masses. We provide a systematic classification of nonlinear stability over the full two-parameter mass plane by combining high-order Birkhoff normal forms, numerical continuation, and singular reduction of resonant normal forms. In nonresonant regions, Arnold's theorem is applied when the quartic nondegeneracy condition holds, while sixth-order normalization resolves the degenerate cases. The $2$:$-1$ and $3$:$-1$ resonances are analyzed using the theorems of Alfriend, Markeev, and Meyer, including a classification of the $3$:$-1$ resonant dynamics. Singular reduction provides the corresponding reduced orbit spaces and reveals saddle-center bifurcations associated with the appearance and disappearance of periodic-orbit families. These results extend previous stability analyses beyond symmetric mass configurations and provide a geometric description of the resonant dynamics in the unequal-mass equilateral restricted four-body problem.