Suppose that $\phi_t$, $t \in \mathbb R$ is a family of diffeomorphisms of a manifold $M$. It is assumed that $\phi_0$ is the identity mapping and dependence on $t$ is smooth, but $\phi_t$ is not a one parameter group: it doesn't hold that $\phi_t \phi_s = \phi_{t+s}$. The question is whether there exists a one parameter family $X_t$ with $X_0=0$ of vector fields such that $\phi_t = \exp ( X_t)$.
I suspect the answer may depend on the type of manifold. For me the two most important choices are compact $M$ and $M = \mathbb R^n$. If the answer to my question is not affirmative in general, I am interested in learning the criteria I need to impose on $\phi_t$ and $M$ to make the statement true.