In this Wiki Link: https://en.wikipedia.org/wiki/Poincaré–Hopf_theorem. It says in the following picture that
"According to the Poincare-Hopf theorem, closed trajectories can encircle two centres and one saddle or one centre, but never just the saddle. (Here for in case of a Hamiltonian system)"
My question is how can we apply the Poincare-Hopf theorem here. The area enclosed by the outer blue curve is a manifold with a boundary. However, the vector field is not always pointing outward along the boundary or the blue curve. How to apply the theorem in this example?
Poincare-Hopf can be also applied to compact manifolds with boundary $M$ and vector fields $X$ which are tangent to the boundary. Assuming that $X$ has no zeroes on $\partial M$, $\chi(M)= ind(X)$. You can reduce this to the case of closed manifolds by gluing to copies of $M$ together along the boundary: The resulting manifold $DM$ has a symmetry fixing the boundary pointwise. Extend $X$ to $DM$ by symmetry. The result is a continuous vector field $DX$ on $DM$ and $$ ind(DX)=2ind(X), \chi(DM)= 2\chi(M). $$ In your case, $M$ is the 2-dimensional disk $B^2$ (bounded by a closed trajectory of $X$) and $\chi(B^2)=1$. Hence, $X$ cannot have a single saddle zero in $B^2$.