Asymptotic analysis of linear periodic systems of homogeneous differential equations with a large or small parameter

The system dot x=(A_0+delta A_1(t))x, A_1in C^+(Bbb R), tin Bbb R^+, with a constant matrix A_0 and T-periodic matrix A_1, is studied. Two cases are considered. par First, if delta=varepsilon is a small parameter, under some assumptions concerning the spectrum of A_0 it is proved that even a more general system dot x=bigl(sum_{k=0}^infty A_k(t)varepsilon^kbigr)x+f(t), x(0,varepsilon)=x^0, can be transformed to the system with almost constant diagonal matrix dot z=Q(t,varepsilon)z+b(t,varepsilon). The result can be viewed as a constructive analogue of the Floquet-Lyapunov theorem. par Second, if delta=varepsilon^{-1} is a large parameter, the singularly perturbed system varepsilon dot x=(sum_{k=0}^infty A_k(t)varepsilon^k)+f(t), x(0,varepsilon)=x^0, is investigated and existence and uniqueness of a uniformly bounded solution on Bbb R^+ is proved for small varepsilon and an asymptotic formula of this solution is given.