Abstract: In the part 2, theorem 3.1 stut ed in part 1^[15] is proved first. The proof is obtained via a way of changing variables to reduce the original system of differentialequations to a form concerning Standard systems of equations in the theory ofdifferentiable dynamical systems. Then by using theorem 3.1 together with thepreliminary theorem 2.l, foe main theorem of this paper announced in part 1 is proved.The definition of admissible perturbation is contained in the appendix of part 2. Themeanings of the main theorem is described in the introduction of part 1.

Key words: vector bundle dynamics, nonuniform hyperbolicity, ergodicity, admissible perturbation, pointwise boundedness