Iterative Sequences of the Localization Method

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