= Smooth flow
{title2=$\phi_t$}
A <smooth flow> on a <smooth manifold> is a smooth map $(t,x)\mapsto\Phi_t(x)$ with
$$
\Phi_0=\operatorname{id},\qquad\Phi_{t+s}=\Phi_t\circ\Phi_s.
$$
The identity implies $\Phi_{-t}=\Phi_t^{-1}$, so each time map is a <diffeomorphism>. Its generating <vector field> is $F(x)=\partial_t\Phi_t(x)|_{t=0}$; trajectories solve the <autonomous differential equation> $\dot x=F(x)$.
For example, $\Phi_t(x)=x+t$ is a smooth flow on the real line, while $\Phi_t(\theta)=\theta+t\pmod{2\pi}$ is a rotation flow on a circle. The second has periodic trajectories, even though the same additive time parameter acts in both examples.
A smooth vector field initially gives only a local flow. The equation $\dot x=x^2$ has $x(t)=x_0/(1-x_0t)$, which blows up at $t=1/x_0$ for $x_0>0$. On a <closed manifold>, compactness prevents escape in finite time and local existence can be continued for every real time. Thus every smooth vector field there generates a global smooth flow. On a noncompact manifold, global existence is a separate property.
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