Localization of solutions of ordinary differential equations near an unstable singular point

A dynamical system is considered in a sufficiently small neighborhood of its nondegenerate Lyapunov unstable equilibrium position. The existence of system trajectories localized in this neighborhood is discussed. An interesting phenomenon of a general nature takes place: when a perturbation is added to the right-hand sides of the equations, the singular point may disappear, but solutions occur that do not leave a small neighborhood of the original singular point.