Perhaps you would be interested in Witten's proof of the Poincare-Hopf theorem. Given a smooth nondegenerate vector field $V$ on a smooth closed manifold $M$, the theorem asserts that the Euler characteristic of $M$ is equal to the sum of the signs of the critical points of $V$. Perhaps this isn't as interesting as the dynamical behavior that you mentioned in your question, but it's a start.
Witten's approach is to use $V$ to perturb the de Rham complex by replacing the de Rham differential $d$ with the operator
$$d_t = d + t i_v \colon\: \Omega^*(M) \to \Omega^*(M)$$
where $t$ is a real number and $i_V$ is the interior product with $V$. He looked at the corresponding perturbed de Rham operator $D_t = d_t + d_t^*$ (where the adjoint is defined using a choice of Riemannian metric) and as usual viewed it as a graded Dirac-type operator on the graded Clifford module $\Omega^*(M)$. $D_t$ is elliptic and hence Fredholm, and since the index of an operator is determined by its symbol class the index of $D_t$ is just the index of the usual de Rham operator $D$ which is the Euler characteristic of $M$.
On the other hand, one can calculate that
$$D_t^2 = D^2 + t^2 ||V||^2 + t T$$
where $T$ is some bundle map. For large values of $t$ the potential term $t^2 ||V||^2$ becomes very large except in a tiny neighborhood of the critical set of $V$, so one can show that the eigenvectors of $D_t$ concentrate near the critical set. Combining this observation with the McKean-Singer formula for the index of $D_t$ and some asymptotic analysis proves the Poincare-Hopf theorem.
There are a variety of generalizations of this result in the literature - perturbing other operators, relaxing the nondegeneracy assumption, etc. I don't know this literature too well and so I don't quite know how much dynamics to expect, but it's worth a look.