Analysis of nonlinear systems by the localization method

In this paper we use localizing sets for compact invariant sets of a nonlinear system to study phase space of the system. It is shown that the localizing set corresponding to a continuously differentiable function delimits simple and complex dynamics of a nonlinear system: the trajectories have a fairly simple behavior outside the localizing set. We prove that the trajectories of the system have the same simple behavior in some regions inside such localizing set.