Resonant phenomena in finite motions of test particles in oscillating dark matter configurations

Nonlinear differential equations are derived that describe the time evolution of the test particle coordinates during finite motions in the gravitational field of oscillating dark matter. It is shown that in the weak field approximation, the radial oscillations of a test particle and oscillations in orbital motion are described by the Hill equation and the nonhomogeneous Hill equation, respectively. In the case of scalar dark matter with a logarithmic self-interaction, these equations are integrated numerically, and the solutions are compared with the corresponding solutions of the original nonlinear system to identify possible resonance effects.