 Solution

Choose a coordinate ball $W$ around $q_1$ and a smaller convex coordinate ball $U$ whose closure lies in $W$. For $q_2\in U$, let $a$ be the constant coordinate vector from $q_1$ to $q_2$. Choose a smooth bump function $\rho$ supported in $W$ and equal to one on $U$, and define in the chartextending it by zero outside $W$. This is a compactly supported smooth vector field. The segment $q_1+ta$ stays in $U$, where $w=a$, so uniqueness for ordinary differential equations givesIn particular, ${\Psi }^1(q_1)=q_2$.