Examples of differential systems with contrasting combinations of Lyapunov, Perron, and upper-limit properties

A number of examples of systems of differential equations are given that have, in a sense, opposite properties of stability or instability of various types: Lyapunov, Perron, and upper-limit. Specifically, all nonzero solutions of one of these systems tend to zero (with unlimited growth of time), nevertheless moving away from it at least once at a specific distance common for all the solutions. In another system, all nonzero solutions starting in a fixed neighborhood of zero tend to infinity in norm, while the other solutions, on the contrary, tend to zero.