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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

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  • $\begingroup$ 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. $\endgroup$
    – Evgeny
    Oct 17, 2013 at 3:56

3 Answers 3

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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.

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NO. Many times stochastic or probabilistic dynamical systems can not be expressed by differential equation. Also many times differential equations are replaced by difference equation for numerical solution or for computer simulation.

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    $\begingroup$ Neither case is in the framework explicitely delimited by the question. $\endgroup$
    – Did
    Oct 16, 2013 at 6:05
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No. Aside from stochastic dynamical systems, mentioned in Supriyo's answer, there are also discrete and hybrid dynamical systems that do not have solutions that are smooth or even continuous. See here for an introduction to hybrid dynamical systems.

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