We consider a general non-autonomous nonlinear small-parameter dynamical system. This system can, for example, represent the differential equations of motion of a dynamical system in the vicinity of stationary motion or equilibrium. In addition to the exact solution, the solutions of the linearized and quasi-linearized system are introduced, in the latter case they have the form of potential series of the small parameter. In the classical definition of Lyapunov stability the perturbation represents solutions for different initial conditions. This definition is not suitable for analyzing the difference between exact and approximate solutions when approximating the problem of motion of real dynamic systems. This is why we apply here Bertolino's generalization 2 of Lyapunov's definition to the different initial condition solutions. The authors do not have information on the application of Bertolino's definition or the problem of the stability of dynamical systems. The sufficient conditions for the stability of a system in the sense of Bertolino's definition. have been formatted.