  In this paper, we establish the sufficient conditions guaranteeing global
uniform exponential stability, or at least global asymptotic stability, of all
solutions for nonlinear dynamical systems, also known as global incremental
stability (GIS) of the systems. We provide here an alternative approach for
assessment of GIS in terms of logarithmic norm under which the stability
becomes a topological notion and also generalize both horizontally and
vertically the well-known Demidovich criterion for GIS of dynamical systems.
Convergence of all solutions to the origin x=0, which is not assumed to be an
equilibrium state of system, is also analyzed. Theory is illustrated by a
simulation experiment.
