Abstract: In this paper we establish the Floguet theory for the quasi-periodic system

where A(u₁,u₂....u_m) is an nxn periodic matrix function ofwith period 2π, and it is of C^τ, τ=(N+ 1)τ₀, τ₀=2(m + 1), N = (1/2)n(n + l).Meanwhile, we define the characteristic exponential roots β₁,β₂,…,β_n of (0.1), and assume that

where K(ω), K(ω,β)->0. k_u, i_v, are integers, all the integers k₁,k₂,…,k_m,k_m are not zero,, ,rhen therequasi-periodic linear transformation, which carries (0.1) into a li near system with constant coefficients.

Key words: turbulence, counter-gradient-transport, TSDIA