Iterative Sequences of the Localization Method

The conditions of positive invariance and compactness of localizing sets and extended localizing sets are proved. The necessary condition for the existence of an attractor in the system is obtained. The concept of an iterative sequence of extended localizing sets is introduced and a condition is obtained under which its elements are positively invariant compact sets and give an estimate of the attraction set. Using the obtained results, the behavior of the trajectories of a three-dimensional system is investigated for acceptable values of its parameters. The conditions of global stability of one of its equilibrium points are found and the set of attraction of another equilibrium point is indicated.