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Abstract Delay coordinates are a widely used technique to pass from observations of a dynamical system to a representation of the dynamical system as an embedding in Euclidean space. Current proofs show that delay coordinates of a given dynamical system result in embeddings generically with respect to the observation function (Sauer et al., 1991). Motivated by applications of the embedding theory, we consider flow along a single periodic orbit where the observation function is fixed but the dynamics is perturbed. For an observation function that is fixed (as a nonzero linear combination of coordinates) and for the special case of periodic solutions, we prove that delay coordinates result in an embedding generically over the space of vector fields in the C r − 1 topology with r ≥ 2. Highlights • Beginning steps in embedology with a fixed observation function. • Proof of genericity of embedding of periodic solutions in R 3. • Neuroscience citations and the Lorenz example are used to show how periodic solutions arise in applications. [ABSTRACT FROM AUTHOR]