On the dynamics of the equilateral restricted four-body problem with Kerr-like primaries

The dynamical behavior of the restricted four-body problem with Kerr-like primaries is investigated numerically. We analyze the positions and linear stability of the equilibrium points and further classify them into index-I and index-II saddle points by evaluating the number of positive eigenvalues of the Hessian matrix of the effective potential. Our results show that the equilibrium points lose linear stability beyond a critical value of the rotation parameter. In addition, we perform a systematic and detailed classification of phase space trajectories. The initial conditions are categorized into three main types: (i) bounded regular orbits, (ii) escaping orbits, and (iii) collision orbits, based on a thorough numerical analysis. Each initial condition is color-coded according to its orbital outcome, yielding visual representations referred to as orbit type diagrams (OTDs). The results reveal a high degree of complexity in the system’s dynamical behavior. In particular, the structure of the escape basins exhibits a strong dependence on the total orbital energy. Fractal basin boundaries are observed across all escape regimes, indicating a complex interplay between the system’s energy levels and the geometry of the escape channels.