Stability in First Approximation

Abstract

In many cases, especially in practical problems, stability of a motion is investigated using equations of first approximation. This is justified not only because of the relative simplicity of the method, but also by the fact that quite often we are able to determine accurately only the first order linear terms that define processes occurring in real systems. However, as was shown in Example 1.1, in investigating stability of a motion, the conclusions arrived at on the basis of equations of first approximation are sometimes absolutely incorrect. Therefore, it is essential to formulate and determine those conditions under which the equations of first approximation will correctly answer the question about the stability of a motion. In general, this problem can be formulated in the following way: Equations of a perturbed motion are given as

$$ \begin{gathered} {{\dot{x}}_1} = {a_{{11}}}{x_1} + \cdots + {a_{{1n}}}{x_n} + {X_1}, \hfill \\ \cdots = \cdots \hfill \\ \dot{x} = {a_{{n1}}}{x_1} + \cdots + {a_{{nn}}}{x_n}, \hfill \\ \end{gathered} $$

((4.1))

where the nonlinear terms X₁, ..., X _n are terms of order higher than one in x₁, ..., x _n (in this chapter we will write just X _k instead of X ^*_k ).