The concept of stability and systems with constant coefficients

Abstract

We shall often consider systems of n first order normal differential equations

$$ {x_{i}}' = {f_{i}}(t,{x_{1}},...,{x_{n}}),\quad i = 1,2,...,n, $$

((1.1.1))

where x^’ _i = dx _i /dt and where t is real and f _i , x _i real or complex. We shall often denote x= (x _i ..., x _n ) as a “point” or a “vector” and t as the “time”, and E _n shall denote the space of the points x. Often (1.1.1) is obtained by a transformation of the m second order Lagrange equations relative to a mechanical system with m degrees of freedom, and thus n = 2m.