Poisson Total Boundedness and Total Oscillability of Solutions to Systems of Differential Equations

Abstract

This is a continuation of the author’s study of some special form of boundedness of solutions to systems of differential equations, namely, their Poisson boundedness. The latter concept generalizes the classical boundedness of a solution and means that there are a ball in the phase space and countably many disjoint intervals on the time half-axis such that the sequence of right endpoints of the intervals tends to the plus infinity and the values of a solution on these intervals lie in the ball. Sufficient conditions for various types of Poisson boundedness of solutions were obtained by using Lyapunov functions, Lyapunov vector-functions, and higher-order derivatives of Lyapunov functions. In particular, there were established some sufficient conditions for the Poisson total boundedness (the Poisson boundedness under small perturbations), the Poisson partial total boundedness, and also the Poisson partial total boundedness of solutions with partially controlled initial conditions. In the present article, we obtain an asymptotic or, in other words, a final characterization of the Poisson boundedness of a solution, which made it possible to establish a connection between the latter concept and an oscillating solution. Further, we introduce the total oscillability of solutions, partial total oscillability of solutions, and partial total oscillability of solutions with partially controlled initial conditions. Using the final characterization of the Poisson boundedness of a solution, and Lyapunov vector-functions with comparison systems, we obtain some sufficient conditions for total oscillability, partial total oscillability, and partial total oscillability of solutions with partially controlled initial conditions. As a consequence, we obtain sufficient conditions for the above types of total oscillability in terms of Lyapunov functions.