Total Poisson boundedness and total oscillability of solutions of systems of differential equations

In the works of the author, the study of a special form of boundedness of solutions of systems of differential equations, namely, their Poisson boundedness, has started. The concept of Poisson boundedness of a solution generalizes the classical concept of boundedness of a solution and means that there is a ball in the phase space and there is a countable system of disjoint intervals on the time semiaxis such that the sequence of right ends of intervals tends to plus infinity and the solution for all values of time from these intervals is contained in the ball. Further, in the author's papers, on the basis of methods of Lyapunov functions, Lyapunov vector functions, and higher-order derivatives of Lyapunov functions, sufficient conditions for various types of Poisson boundedness of all solutions were obtained. In particular, sufficient conditions were obtained for total Poisson boundedness (Poisson boundedness under small perturbations), partial total Poisson boundedness, and also partial total Poisson boundedness of solutions with partially controlled initial conditions. In this paper, we obtaine an asymptotic or, in other words, final characterization of the concept of Poisson boundedness of a solution, which made it possible to establish a connection between the concept of a Poisson bounded solution and the concept of an oscillating solution. Further, the concepts of total oscillating of solutions, partial total oscillating of solutions, and partial total oscillating of solutions with partially controlled initial conditions are introduced. Based on the above final characterization of the concept of Poisson boundedness of a solution, and also on the basis of the method of Lyapunov vector functions with comparison systems, we obtain sufficient conditions for total oscillating, partial total oscillating, and partial total oscillating of solutions with partially controlled initial conditions. As a consequence, sufficient conditions for the above types of total oscillating of solutions are obtained in terms of Lyapunov functions.