On exponentially attracting invariant manifolds of ODEs

Let M be an invariant manifold of an autonomous system of ODEs. We discuss the question: when do all trajectories from some neighbourhood of M quickly tend to M? The linearized equations are considered for solutions of the initial system with trajectories lying on M. Assuming that M is smooth and compact, the following theorem is proved. Them manifold M is exponentially attracting if and only if all components, transversal to M, of solutions of the linearized equations exponentially tend to zero.