The stability of the equilibrium of a conservative system

In this paper, we consider the conservative system x+a(t)x(2n+1)+e(t,x)=0, n greater than or equal to 1, where a(t) is a continuous and 1-periodic function in the time t. e(t,x) is also 1-periodic in the time t and dominated by the power x(2n+2) in a neighborhood of x = 0. A sufficient and necessary condition for the stability in the Liapunov sense of the trivial solution x = 0 is given. (C) 1996 Academic Press, Inc.