Suppose $v$ is a (possibly time-dependent) vector field on a compact manifold $M$.


Its flow is a mapping $g: M \times \mathbb{R} \rightarrow M$, where $g$ satisfies the following conditions (written in the local chart):


$g(x, 0) = x$


$\frac{dg}{dt} (x, t) = v(g(x, t), t)$


I would like to find for a fixed $\epsilon$ such $\delta$ that, for any $\delta$-variation of the vector field, the differential flow for time $1$ would not perturb more than on $\epsilon$.


The norm on vector fields is $\sup\_{x\in M}(|v(x)|)$ w.r.t. to some arbitrary riemannian metric on $M$, and the distance between mappings $f$, $g$ is defined as $\sup\_{x \in M}(dist(f(x), g(x)))$.


While this fact is almost surely standard, I can not figure out if it follows from the continuous dependence of the solution of ODE on parameters. Does anyone know a good reference?



Continuity of the differential flow under a perturbation of the vector field
Let me consider two autonomous vector fields $X\_1, X\_2$ on a compact smooth manifold $\mathcal M$ and assume that the Lipschitz condition is true for both of them. The flow $\psi\_j$
of $X\_j$ is defined on $\mathbb R\times \mathcal M$
by
$$
\dot \psi\_j(t,m)=X\_j(\psi\_j(t,m)), \quad \psi\_j(0,m)=m.
$$
Now we consider for $m$ in a coordinate chart and $t\ge 0$ small enough
$$
\psi\_2(t,m)-\psi\_1(t,m)=\int\_0^t\bigl(
X\_2(\psi\_2(s,m))-X\_1(\psi\_1(s,m))\bigr) ds,
$$
so that
\begin{multline}
\vert\psi\_2(t,m)-\psi\_1(t,m)\vert\le 
\int\_0^t\bigl\vert
X\_2(\psi\_2(s,m))-X\_1(\psi\_1(s,m))\bigr\vert ds
\\
\le 
\int\_0^t\bigl\vert
X\_2(\psi\_2(s,m))-X\_2(\psi\_1(s,m))\bigr\vert ds
+\int\_0^t\bigl\vert
X\_2(\psi\_1(s,m))-X\_1(\psi\_1(s,m))\bigr\vert ds
\\
\le 
\int\_0^t
L\_2\vert\psi\_2(s,m)-\psi\_1(s,m)\vert ds
+t\Vert X\_2-X\_1\Vert=R(t).
\end{multline}
We have thus with $\rho(t)=\vert\psi\_2(t,m)-\psi\_1(t,m)\vert$,
$$
\dot R(t)=L\_2\rho(t)+\Vert X\_2-X\_1\Vert\le L\_2 R(t)+\Vert X\_2-X\_1\Vert,
$$
so that 
$
\frac{d}{dt}\log\bigl(L\_2 R(t)+\Vert X\_2-X\_1\Vert\bigr)\le L\_2
$
and
$$
\log\bigl(L\_2 R(t)+\Vert X\_2-X\_1\Vert\bigr)\le \log\bigl(\Vert X\_2-X\_1\Vert\bigr)+ L\_2 t,
$$
entailing
$L\_2 \rho(t)+\Vert X\_2-X\_1\Vert\le 
L\_2 R(t)+\Vert X\_2-X\_1\Vert\le e^{L\_2 t}\Vert X\_2-X\_1\Vert
$
and 
$$
\rho(t)\le L\_2^{-1}(e^{L\_2 t}-1)\Vert X\_2-X\_1\Vert.
$$


