Non-linear stability of the equilibrium of a system of mass points

By applying the non-stationary tensor virial theorem to spheroidal systems of mass points with an anisotropic velocity distribution, we have found that, when the total energy is negative, there is a unique equilibrium for a given such system. In this paper, we discuss the non-linear stability of the equilibrium. We find that, in the neighborhood of this equilibrium, spheroids execute quasi-periodic or near-quasi-periodic oscillations. Though chaoticity and orbit diffusion are present in some cases, the equilibrium is practically stable with a large stable region. This result would be useful in n-body simulations aimed for a better understanding of stellar system oscillation.