Lyapunov Vector Functions, Krasnosel'skii Canonical Domains, and Existence of Poisson Bounded Solutions

We introduce the concepts of Poisson boundedness and partial Poisson boundedness for a solution of a differential system. These properties mean that the solution or, respectively, its projection onto a given subspace is contained in some ball for the values of an independent variable belonging to countably many intervals converging to infinity. Based on the method of Lyapunov vector functions and the Krasnosel'skii canonical domain method, sufficient conditions are obtained for the existence of such solutions.