Invariant Foliations for Carath Eodory Type Diierential Equations in Banach Spaces

Diierential equations which explicitly but discontinuously depend on time are rarely studied objects even though they promise important applications e.g. in control theory or in the theory of random dynamical systems. In this paper we continue a previous study of qualitative properties of so-called Carath eodory type diierential equations whose feature is measurable dependence of the right-hand side on time. In fact, we show that the fundamental theorem on the existence of integral manifolds can be generalized to a result providing two complete foliations of the extended state space by integral manifolds. This detailed information about the dynamical structure of the extended state space, on the other hand, can be used to construct transformations establishing the topological equivalence between certain weakly coupled and completely decoupled systems.