I would like to know an example of a compact Riemannian manifold $M$ and a smooth vector field $X$ on $M$ such that the flow $F$ associated to $X$ is defined for all time and for some vector $v\in T\_pM$, the norm $$|dF\_t(v)|\rightarrow \infty$$ as $t\rightarrow\infty$. Thanks. 



An example where the norm of the differential of a flow grows unboundedly
For a minimal example, take $M:=\mathbb{S}^1\times \mathbb{S}^1$ where $\mathbb{S}^1:=\mathbb{R}/2\pi \mathbb{Z},$ with the field $X(u,v):=\sin(u)\partial\_v.$ Then $F\\ ^t(u,v)=\big(u,v+\sin(u)t\big)$ and the differential of the flow at time $t$, computed on the vector $\partial\_u$ at $p=(0,0)$ is just $\partial\_u+t\partial\_v.$


