Sufficient roughness conditions of non-autonomous nonlinear dynamic system in the sense of stability type conservation

The conditions under which the stability type of the equilibrium state of a Lyapunov-type random-order non-autonomous nonlinear dynamic system would not change under any relatively small linear or nonlinear perturbances of its right-hand member (roughness in the sense of stability type conservation) are considered. Nonlinear components of right-hand members of unperturbated system and nonlinear perturbances of those right-hand members are considered to belong to a wide class of nonlinear functions depending in general case not just on the phase variable but also on time. This class contains both analytic functions and various types of nonanalytic functions. Sufficient roughness conditions are derived.