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?