Suppose that $F$ is a $C^\infty$ diffeomorphism of the plane with hyperbolic fixed point $p$ for which a branch of the unstable manifold, $W^u_+(p)$, has a same-sided quadratic tangency with the stable manifold, $W^s(p)$. If the eigenvalues of $DF$ at $p$ satisfy a non-resonance condition, each nonempty open set of $ \cl( W^u_+(p))$ contains a copy of any continuum that can be written as the inverse limit space of a sequence of unimodal bonding maps.