ABSTRACT
This chapter focuses on the stability analysis of solutions of systems of differential equations involving small parameters. The problem is solved in terms of a new generalization of A. M. Lyapunov’s direct method for investigation of the stability of motion. This generalization is developed for the following types of systems: large-scale systems with weakly interacting subsystems; oscillating systems; systems with nonasymptotically stable unperturbed systems; and systems that can be investigated via perturbed Lyapunov functions only. The systems of differential equations are applicable in the theory of nonlinear oscillations to mechanics, physics, electronics, electrical engineering and biology. The chapter presents results obtained by combining the averaging method and the method of Lyapunov functions on the basis of limiting equations. It also investigates time intervals for which the solution does not leave some sets that are time-variable in the general case.
