Embeddability of time scale dynamics in ode dynamics

In this paper we consider dynamical equations x(Delta) = f (t, x) on time s cales T. Here T is an arbitrary closed subset of R, x(Delta) denotes the di fferential-difference. operator according to T. In the context of the quest ion, whether the dynamical system phi : T x T x X --> X, generated by the a bove equation, is a restriction of an ordinary continuous dynamical system Phi : R x R x X --> X, generated by an ordinary differential equation, we s tudy appropriate conditions and counterexamples. Our main tools are general topological principles, Hermite interpolation and a certain version of cha in rule for time scales.