Let $p$ be a saddle point of the planar ODE $x' = f(x)$ with $f$ smooth. Suppose $\gamma$ is a homoclinic orbit starting and ending in $p$.
Define $\Gamma := \gamma \cup p$ and let $\mathcal{U}$ be the open region enclosed by $\Gamma$.
I would like to prove that $\mathcal{U}$ containts either an equilibrium point or a periodic orbit.
I thought about using the Poincaré-Bendixson theorem but I'm having trouble making it work.
I know $\Gamma \cup \mathcal{U}$ is a trapping region, but it contains the fixed point $p$. On the other hand, if I exclude $p$, the region is not closed anymore.
Am I looking in the right direction?
Edit: I am actually starting to doubt if this is even true. Isn't it possible that all trajectories starting in $\mathcal{U}$ end up in $p$?