The stability of an nth-order nonlinear time-varying differential system

The stability of a system described by an
n
th order differential equation
y^{(n)} + a_{n-1}y^{(n-1)} + . . . + a_{1}y + a_{0} = 0
where
a_{i}=a_{i}(t, y, \dot{y}, . . . , y^{(n-1)}), i=0, 1, . . . , n - 1
, is considered. It is shown that if the roots of the characteristic equation of the system are always contained in a circle on the complex plane with center
(-z, 0), z > 0
, and radius Ω such that
\frac{z}{\Omega} > 1 + nC_{[n/2]}
where
[n/2]
= nearest integer
\geq n/2
and
nC_{m} = n!/m!(n-m)!
, where
n
and
m
are integers, then the system is uniformly asymptotically stable in the sense of Liapunov.