On invariant manifolds in singularly perturbed systems

A singularly perturbed system with a small parameter e at the velocity of the slow variable y and with the fast variable x is considered. The main hypothesis is that for all y from some bounded domain D, the fast subsystem has a stable invariant or overflowing manifold M0(y) and that the motions in this system going in the directions transversal to M0(y) are more fast than the mutual approaching of trajectories on M0(y) (a precise statement is given in terms of appropriate Lyapunov-type characteristic numbers). It is proved that for a sufficiently small ε, the whole system has an invariant manifold close to ∪yεD M0(y) × {y}; the degree of its smoothness is specifed. © 1999 Kluwer Academic/Plenum Publishers.