Stability Analysis

This monograph describes the results of stability investigations of solutions of five types of nonlinear ordinary differential equations containing a small parameter. The averaging technique, the comparison method and Lyapunov's direct method for scalar, vector or matrix functions are employed for the construction of various sufficient conditions for the stability of solutions. Proofs of various theorems are presented including the averaging theorems of finite and unbounded intervals of standard systems of differential equations using the comparison method. Inverse theorems are proved for the stability theorems, and thus the universality of the comparison method with a scalar Lyapunov function for the given class of systems is shown. Systems describing motion in the form of smooth drift and fast oscillations are investigated using the averaging principle, which permits evolution equations that contain only slow variables to be obtained. Supplementing this approach with the comparison method, new averaging theorems are proved for systems with fast and slow variables on finite and unbounded intervals.
The investigation of the stability of singularly perturbed large-scale systems based on the application of a Lyapunov matrix function is also presented as well as algebraic conditions for the stability of solutions of large-scale systems based on the appropriate Lyapunov matrix function.