Lyapunov, Perron and upper-limit stability properties of autonomous differential systems

For a singular point of an autonomous differential system, the natural concepts of its Perron and upper-limit stability are defined, reminiscent of Lyapunov stability. Numerous varieties of them are introduced: from asymptotic and global stability to complete and total instability. Their logical connections with each other are investigated: cases of their coincidence are revealed and examples of their possible differences are given. The invariance of most of these properties with respect to the narrowing of the phase region of the system is established.