Inertial Flows, Slow Flows, and Combinatorial Identities for Delay Equations

abstract

The vector field induced on the finite-dimensional inertial manifold of a delay equation with small delay is proved to agree, up to the order of the expansion, with the vector field induced on a slow manifold of the differential equation obtained from the delay equation by expanding to some finite order in powers of the delay. In addition, the smoothness of inertial vector fields, the smoothness of slow vector fields, and the existence of combinatorial-style identities obtained by equating the series expansions of the slow and inertial vector fields are discussed.