# are all dynamical systems described by differential equations?

we defined in lecture a dynamical System as a one-parameter family of maps $\phi^t:M\rightarrow M$ such that $\phi^{t+s}=\phi^t\circ\phi^s$ and $\phi^0=Id$, where $M$ is some (smooth) manifold and $s,t\in (a,b)\subset\mathbb{R}$.

Of course, if we consider some vector field $V:M\rightarrow TM$, then the flow of that vector field around some point $x_0\in M$ is a (local) dynamical system.

Now I'm wondering if all dynamical systems can be described that way. Can we find for all dynamical systems $\phi^t$ a vector field $V$, s.t. $\phi^t$ is the flow of $V$? Maybe you know some argument.

Regards

• Also, be careful: picking just some vector field $V: M \to TM$ doesn't guarantee that you'll get a flow. There are examples when the only thing that can be obtained is a semiflow. Oct 17, 2013 at 3:56

Assuming differentiability, define a vector field $V(t)$ by the differential of $\phi^t$, i.e. $V(t)=D\phi^t$. The ($t$-dependent) flow of $V(t)$ is what you're looking for.