ИСТИНА

We discuss a new behavior of a dynamic system near an unstable equilibrium position. We call corresponding trajectories located in the selected neighborhood of unstable equilibrium as localized trajectories. The use of topological methods for proving the existence of localized trajectories makes possible to abandon the condition of analyticity of the first integrals and the condition of non-resonance of purely imaginary roots of the characteristic equation for the systems of Lyapunov. As an important application, we consider perturbed motion in libration points vicinity of the restricted circular three-body problem. Numerical simulations for the parameters of the Earth-Moon system convincingly illustrate our theoretical study.