Definition and properties of measures of stability and instability of zero solution of a differential system (Q6114321)

In the framework of finite-dimensional ordinary differential equations \[ \dot x=f(t,x) \] the author introduces so-called measures of stability (and instability). These real numbers in \((0,1]\) are intended to provide estimates from below on the possibility of choosing an initial value that is arbitrarily close to zero, so that the corresponding solution curve stays in a given neighborhood of the trivial solution in any of the following senses: (a) on the entire positive semiaxis, (b) at least sporadically, but at arbitrarily large instants and, (c) eventually, i.e.\ beginning from some positive instant but then forever. Characterizations and properties of these (in)stability measures are given. \par As special cases scalar, autonomous and linearly-homogeneous equations are considered, as well as planar systems.