Existence of Localized Motions in the Vicinity of an Unstable Equilibrium Position

We consider a dynamical system whose equilibrium position is nondegenerate and Lyapunov unstable, the degree of instability being greater than zero and less than the number of degrees of freedom. We show that for any sufficiently small positive value of the total energy of the system, there exists a motion of the system with this energy that starts at the boundary of the region of possible motion and does not leave a small neighborhood of the equilibrium position. Such motions are called localized motions.