TITLE: Links between difference and differential equations?
QUESTION [45 upvotes]: Does there exist any correspondence between difference equations and differential equations?
In particular, can one cast some classes of ODEs into difference equations or vice versa?

REPLY [2 votes]: First see the correspondence between discrete and continuous dynamical systems. Both are acts of a monoid on some set $X$.
For discrete dynamical systems the monoid that acts is $\mathbb{N}$. For continuous dynamical systems the monoid that acts is $\mathbb{R}$.
This action comes in the form of a left multiplication 
$(.,.):\mathbb{N}\times X \rightarrow X$ or $(.,.):\mathbb{R}\times X \rightarrow X$
and is compatible with the addition structure of $\mathbb{N}$ and $\mathbb{R}$ respectively. 
I.e. for all $n,m \in \mathbb{N}$ or $\mathbb{R}$ and $x \in X$ we have $(0,x) = x$ and $(n,(m,x)) = (n + m,x)$.
Now on the story of difference and differential equations. A first order difference equation equals a discrete dynamical system. Note that any difference equation can be converted to a system of first order difference equations (see higher order difference equations). Hence any difference equation equals a discrete dynamical system. Note that the $\mathbb{N}$-act is given by $(n,x) = f^n(x)$.
If the differential condition is sufficiently smooth there exists a unique solution for any point ($\phi_x^t$). We use this solution as the $\mathbb{R}$-act. I.e. $(t,x) = \phi_x^t$. Hence any sufficiently smooth differential equation equals a continuous dynamical system.