Let $Q^n$ be a closed, Riemannian manifold, $TQ$ its tangent bundle with the canonical lift of the Riemannian metric (as outlined in do Carmo) and the resulting compatible triple, $\xi$ a complete tangent vector field on $TQ$, and $p$ [an elementary, in the notation of Abraham-Robbin] a hyperbolic [in the rest of the universe's notation, evidently] fixed point for $\xi$. Let $W^s(p)$ and $W^u(p)$ be the stable and unstable manifolds of $p$ respectively, and suppose both are closed. If both $W^s$ and $W^u$ are orientable, orient $TQ$, $W^s$, and $W^u$ ($TQ$ is always orientable); in any case, adjust $W^s$ and $W^u$ so they are transverse to each other (although each may itself be only immersed). Let $\{q_1, q_2, \ldots, q_k\} = W^{s\prime } \cap W^{u\prime}$.
Is there any relevant information relating to $\xi$ that can be determined from the mod 2 intersection number (if either or both of $W^s$ or $W^u$ is non-orientable), integral intersection number (if both $W^s$ and $W^u$ are orientable), or integral group-ring intersection number (if $\pi_1(TQ,p)$ is known) of $W^s$ and $W^u$?
(Of course, if one only has a coordinate patch for $TQ$ with generalized (resp. symplectic) coordinates which excludes a set of measure 0 and a Lagrangian (resp. Hamiltonian) on the coordinate patch, neither the fundamental group of $TQ$ nor the fundamental group elements determined by the homoclinic loops at $p$ determined by each $q_i$ may be known, and so we may not be able to compute the $\mathbb{Z}[\pi_1(TQ,p)]$ intersection number of $W^s$ and $W^u$; the mod 2 or integral intersection number should still be computable without determining fundamental group of $TQ$, depending on whether $W^s$ and $W^u$ are both orientable.)
[Edit: According to Abraham-Marsden, hyperbolic is now the standard terminology.]
This is probably the application: if the intersection number is 0, the submanifolds can be made disjoint by The Whitney Trick (See this post). #NoDepositNoReturn