Parametric stability analysis for planar bicircular restricted four-body problem

Stability for the non-autonomous bicircular four-body model is analytically investigated in this study. The governing equation is derived from Newton’s law of gravity. When the distance between the infinitesimal mass and the third primary is expanded as Taylor expansions, the governing equation can be regarded as two parts: the unperturbed conservative system and the small periodically parametric excitations. The unperturbed system’s natural frequency and parametric frequency are analyzed for the possibility of principal parametric resonances. The method of multiple scales is applied directly to the governing equation. The stability conditions are obtained analytically for the principal parametric resonance. Numerical method demonstrates the efficiency of the analytical results.