STUDY OF STABILITY IN A FIRST APPROXIMATION

Abstract

This article examines first-approximation methods for studying the stability of dynamic systems. The focus is on two key approaches: the linearization method and the Lyapunov method. The linearization method approximates a nonlinear system near an equilibrium point using its linear model, which significantly simplifies the stability analysis, especially when analytical solutions are available. However, this method is applicable only when the linearized system provides an accurate representation of the original system’s behavior. The Lyapunov method, on the other hand, is a more powerful tool for analyzing stability since it allows stability assessment without explicitly solving the system's equations. This article provides a detailed discussion of the fundamental principles of constructing Lyapunov functions, stability criteria, and examples of their application to various classes of dynamic systems. Additionally, concrete examples are presented to illustrate the practical applications of these methods in mechanics, control theory, and mathematical physics. The limitations of these methods and possible ways to overcome them are also discussed. This article is intended for students, researchers, and specialists working on the stability analysis of dynamic processes in various scientific and engineering fields.