Let $M$ be a compact manifold and $\varphi : M \longrightarrow M$ be a diffeomorphism which is isotopic to the identity. Does there exist a vector field $ X $ on $M$ such that $\varphi$ is the flow at time $1$ of $X$? If that is not always the case, where does the obstruction for such a $\varphi$ to be a flow lives in ?



Flows of vector fields and diffeomorphisms isotopic to the identity
The answer is no. In fact, there are diffeomorphisms arbitrarily close to the identity which are not contained in flows (which are also often called $1$-parameter subgroups; here one is thinking of the set of vector fields on $M$ as a sort of "Lie algebra" for the diffeomorphism group).


Here's an example which I learned about from Warning 1.6 of Milnor's paper "Remarks on infinite-dimensional Lie groups" (which I highly recommend reading if you are interested in things like this). Regard $S^1$ as $\mathbb{R}/2\pi$. Fix some integer $n$ and some $\epsilon$ such that $0<\epsilon<1/n$. Define
 $$f : S^1 \longrightarrow S^1$$
 $$f(x) = x + \pi/n + \epsilon \sin^2(n x).$$
Choosing $n$ large enough and $\epsilon$ small enough, we can make $f$ arbitrarly close to the identity in the $C^{\infty}$
topology. It is a fun exercise to show that $f$ does not lie in any $1$-parameter subgroup of $\text{Diff}(S^1)$.


It is not hard to generalize this kind of behavior to any manifold.


I don't think there is any easy description of the obstruction here; it seems to be a delicate problem in dynamics.


However, it is the case that every element of $\text{Diff}^0(M)$ can be written as a product of finitely many elements that are contained in $1$-parameter subgroups. Indeed, the set of elements of $\text{Diff}^0(M)$ that are contained in $1$-parameter subgroups generate a subgroup $\Gamma$ that is easily seen to be normal. But Thurston proved that $\text{Diff}^0(M)$ is simple for any compact manifold $M$, so we must have $\Gamma = \text{Diff}^0(M)$. For a discussion and proof of Thurston's theorem, I recommend Banyaga's book "The Structure of Classical Diffeomorphism Groups".


