A spectral method for studying the stability of some classes of nonautonomous differential equations

The author proposes a method to investigate (without using Lyapunov functions) the stability of the trivial solution of differential systems of three types: (1) dot x=A(t,epsilon)x+f(x,t), x(0,epsilon)=x^0, where A(t,epsilon)=sum^infty_{k=0}A_k(t)epsilon^k, f(0,t)equiv 0, |A(t,epsilon)|leq C, tgeq 0, |epsilon|leqepsilon_0<1; (2) epsilondot x=A(t,epsilon)x+epsilon b(x,t), x(0,epsilon)=x^0, where the series A(t,epsilon)=sum^infty_{k=0}A_k(t)epsilon^k is absolutely and uniformly convergent for tgeq 0, |epsilon|leqepsilon_0; (3) dot x=A(t)x+f(x,t), x(t_0)=x^0, where A(t)=t^msum^infty_{k=0}A_k(t)t^{-k}, mgeq 1, f(0,t)equiv 0, tgeq t_0geq 1. (For all systems the A_k are sufficiently smooth T-periodic matrix functions.) par Conditions for stability, asymptotic stability and instability are obtained. Some examples are given to illustrate the proposed methods.