On The Reducibility Of Linear Differential Equations With Quasiperiodic Coefficients Depending On Parameters

This paper treats the reducibility of system of quasiperiodic linear differential equation x|+ = (A(ξ) + Q(t, ξ)x,x∈ R~n with a parameter ξ, where .A(ξ) is a constant matrix, Q(t,ξ) is a quasiperiodic matrix depending on ξ. Suppose that the frequency parameter of Q(t,ξ) satisfies the Rüssmann non-degenerate condition, and some non-resonant conditions with eigenvalues of A(ξ). First, we prove that when Q(t,ξ) is sufficient small for most ξ in the sense of L-measure, if A(ξ) has n different eigenvalues , the system is reducible. Second ,we prove that if A(ξ) has multiple eigenvalues ,the system is reducible,too.