Chaotic dynamics of a test particle around a gravitational field with a dipole

In this paper, we investigate the dynamics of a test particle in a gravitational field with a dipole. First we study the gravitational potential by numerical analysis, and find that, for some appropriate parameters, there are two different cases in the potential curve. One is a one-well case with a stable critical point, and the other is a three-well case with three stable critical points and two unstable critical points. We also investigate the evolution of the test particle in the phase space. By performing Poincaré sections for different values of the parameters and initial conditions, we find a regular motion and a chaotic motion. From these Poincaré sections, we further confirm that the chaotic motion of the test particle mainly originates from the dipole.