Approximate a Discrete time dynamical system by a continuous one

Question

Is there any method by which we can somehow "embed" a non linear discrete time system into a continuous time dynamical system? (Assume discrete time system here is a set of non linear difference equations and a continuous time system is a set of differential equations) I know the reverse is quite easy but I cant seem to find methods by which I can approximate a discrete time system by a continuous one.

Answer 1

Not sure if this answers your question but if you want to capture the global dynamics I would first look into constructing a Morse decomposition of the dynamical system at hand. Morse decompositions are specifically interesting since these capture how the invariant sets are connected to each other by constructing an ordering on them. The discrete Morse decomposition might have a continuous analogue from which you might deduce the overall dynamics of the continuous system.

For the local dynamics you can just use flow boxes to reconstruct a continuous vector field from the discrete

Answer 2

In Palis/de Melo's "Geometric theory of Dynamical Systems" there's a notion of "suspension of diffeomorphism": a way to construct vector field from a diffeomorphism of compact manifold $M$ embedding it as Poincare section for the flow. Not sure that it's exactly can be applied to your case, but nevertheless can give you an idea of solution.