This paper is primarily concerned with linear time-varying ordinary differential equations. Sufficient conditions are given for the existence of an exponential dichotomy or equivalently an invariant splitting. The conditions are more general than those given in Part I of this paper and include the case in which the coefficients lie in a base space which is chain-recurrent under the translation flow and also the case in which compatible splittings are known to exist over invariant subsets of the base space. When the compatibility fails, the flow in the base space is shown to exhibit a gradient-like structure with attractors and repellers. Sufficient conditions are given guaranteeing the existence of bounded solutions of a linear system. The problem is treated in the unified setting of a skew-product dynamical system and the results apply to discrete systems including those generated by diffeomorphisms of manifolds. Sufficient conditions are given for a diffeomorphism to be an Anosov diffeomorphism.
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In an earlier paper  we began an investigation into the question of * the existence of exponential dichotomies for linear differential equations with time-varying coefficients. This paper is both a sequel and an amplification of this earlier work. The dynamical object studied in  was a linear skew-product flow on a product space X x Y and its generalization, the linear fiber-preserving * This research was begun while the first author was visiting the University of Minnesota. Robert J. Sacker was partially supported by U. S. Army Contract DAHC/ 04-74-6-0013 and George R. Sell by NSF Grant No. GP-38955.