Embedding discrete flows on R in a continuous flow

The problem of determining when a given discrete flow on a topological space is embeddable in some continuous flow was mentioned by G. R. Sell (“Topological Dynamics and Ordinary Differential Equations,” Van Nostrand, New York, 1971) in his book on topological dynamics. In this book, the theory of generalized dynamical systems is exploited in the qualitative study of differential equations. Even more complicated is the problem of simultaneously embedding two or more discrete flows in a single continuous flow. We examine both of these problems when the underlying topological space is the space R of the real numbers.