Equivalence of discrete and continuous dynamical systems?

Question

I know that the flow of a continuous dynamical system can be viewed as a map describing the correspondent discrete dynamical system. Reversely, can the latter be used to define its continuous counterpart?

Answer 1

You can pass from the continuous system to the discrete one because you are actually restricting the flow to a subset of the set in which time is defined previously. Let's say that $\phi : \mathbb{R}\times\mathcal{Q}\rightarrow\mathcal{Q}$ is the continuous flow. Then you can say that the associated discrete one is associated to integer time instants, so actually you define the discrete trajectory of $x_0\in\mathcal{Q}$ as the following infinite sequence: $\{\cdots,\hat{\phi_{0}}(x_0),\hat{\phi_1}(x_0),\cdots\}$ where $\hat{\phi_n}(x_0) := \phi(t=n,x_0)\;\forall\,n\in\mathbb{Z}$.

So you are just "selecting" some time instants for each trajectory and hence you define a discretization of the continuous paths made by the continuous system.

The problem going from the discrete flow to the continuous one is that to "join" each contiguous couple of points of the discrete trajectories is not an operation you can do in a unique way. So for instance for each discrete flow/trajectory you can define arbitrarily many continuous associated flows.