For a smooth flow \(F:M\times\mathbb R\to M\) on a smooth manifold \(M\) let \(\mathcal E(F)\) consist of those \(C^\infty\) maps \(M\to M\) which map each orbit into itself and which are local diffeomorphisms at every singular point of \(F\). Let \(\mathcal D(M)\) be the group of \(C^\infty\) diffeomorphisms and \(\mathcal D(F)=\mathcal E(F)\cap\mathcal D(M)\). It is shown for a class of flows that the inclusion \(\mathcal D_{\text{id}}(F)\subset\mathcal E_{\text{id}}(F)\) is a homotopy equivalence and both spaces are either contractible or homotopy equivalent to \(\mathbb S^1\) when the Whitney W\(^r\) topology (\(0\leq r\leq\infty\)) is used, where the subscript id means the identity component.