On the existence of a globally Lyapunov unstable differential system all of whose solutions tend to zero as time tends to infinity (Q2082813)

This paper is devoted to a classical problem in the stability theory of differential equations, namely the convergence of all solutions to zero does not imply the Lyapunov stability. The author has constructed a two-dimensional nonautonomous system whose solutions tend to zero as time tends to infinity, but each of the nonzero solutions always moves sufficiently far away from zero before going close to zero. In addition, this system has zero first approximation. Though the answer to the problem is well known, the construction as well as the proof is not trivial. The author also recalls the notions of Perron stability and the upper-limit stability. Alternatively speaking, the system given in this paper has global Perron and upper-limit stability, but it is globally Lyapunov unstable. It is noted that one cannot construct similar examples for neither autonomous systems nor one-dimensional ones.