Existence of localized movements in the vicinity of an unstable equilibrium position

The paper considers a dynamic system whose equilibrium position is nondegenerate and unstable according to Lyapunov, and its degree of instability is greater than zero and less than the number of degrees of freedom. It is shown that for any sufficiently small positive value of the total energy of the system, there is a motion of the system with a given energy value that begins at the boundary of the region of possible motion and does not leave a small neighborhood of the equilibrium position. We call such motions localized.