Can a hyperbolic fixed point be inside a homoclinic loop on a plane? Consider a system of odes on a plane. It is obvious that a fixed point of centre-type stability (imaginary e-values) can exist inside a homoclinic loop to a hyperbolic saddle fixed point. For example, take the Hamiltonian $H(x,y) = \frac{1}{2}(y^{2}-x^{2}) + x^{3}$. Here the hyperbolic saddle point at origin has a "fish" shaped homoclinic loop enclosing the centre fixed point at $(1/3, 0)$.
What I'm wondering is whether a homoclinic loop can enclose fixed points other than centre type, i.e. can a "sink / source" or a "saddle" exist inside a homoclinic loop? And can this situation be generalised to $\mathbb{R}^{n}$?
I cannot think of a way to either prove / disprove this. I suspect Poincare-Bendixson and index theory could be used, but not sure how.
 A: Let me address the index sum issue which came up in the comments, in order that the example described in the first comment of @Evgeny can be tailored to work.
For a planar flow generated by a vector field $\vec v$, with a homoclinic loop $\ell$ based at a saddle point $p$, and letting $D$ be the 2-disc bounded by $\ell$, one can apply index theory indirectly to conclude that the index sum over the singularities of $\vec v$ in the interior of $D$ must equal $+1$. And then one can construct examples where there is a saddle singularity on the inside, although of course that saddle singularity must be balanced appropriately with other singularities, including at least one of positive index, so that the index sum equals $+1$. For example, one saddle and two center singularities can be obtained (I'll say more about this below).
To carry out the index theory argument, one chooses a small enough neighborhood $U$ of $p$ which is disjoint from all the zeroes of $\vec v$ except $p$ itself, and one then simultaneously perturbs $D$ and $\vec v \mid D$ near $U$ to obtain a new disc $D'$ and new vector field $\vec v'$ on $D'$, so that the boundary of $D'$ is smooth, $\vec v'$ is tangent to the boundary of $D'$, $\vec v'$ has no zeroes in $U \cap D'$, and outside of $U$ we have $D-U=D'-U$, $\vec v \mid D-U = \vec v' \mid D'-U$. Thus the sum of the indices of the zeroes of $\vec v$ in $D$ equals the sum of the indices of the zeroes of $\vec v'$ in $D'$ which equals $+1$.
And then you can work backwards. Start, for example, from the example given in the link of the first comment of @Evgeny, which is defined on a nice smooth round disc $D'$ with 2 centers and 1 saddle (index sum $+1$). In this example the vector field $\vec v$ points inward along the boundary of $D'$. So, first one will have to alter that example so that the vector fields points tangentially along the boundary of $D'$; one can, for example rotate the vector field along and near the oundary of $D'$ to achieve this purpose. After that, one can do a further alteration, "unperturbing" the example near a point on the boundary, to create a saddle singularity and homoclinic loop.
