# Difference between an ODE and dynamical system?

I don’t really understand the difference between an ODE and a dynamical system. The ODE just seems like a way to describe it. Is there more to it?

Are there dynamical systems which cannot be represented by any ODE?

• You haven't mentioned few quite important things. First, there is rigorous definition of dynamical system beyound words "state space" and "evolution operator". Second, when set of times is $\mathbb{R}$ and state space is some manifold or Euiclidean space, dynamical system corresponds to an ODE. Third, not all ODEs define a proper dynamical system. There are two obstructions usually: non-uniqueness of solutions (non-existent for smooth ODEs) and finite-time .... – Evgeny Jan 25 '18 at 15:23
• ... blow-up (when trajectories doesn't exist for all $t \in \mathbb{R}$, like in $y' = y^2$). The latter one can be fixed by some transformation that takes trajectories of one ODE to another, but trajectories of the second ODE are defined for all times. So, flows in Euclidean space or manifolds (as a dynamical systems) and ODEs are indistinguishable, they are the same. ODEs are not the one of the ways to describe particular flow. Of course I agree that ODEs are a particular example of dynamical system beside many others. – Evgeny Jan 25 '18 at 15:25