A point where a stable and an unstable separatrix (invariant manifold) from the same fixed point or same
family intersect. Therefore, the limits
and
exist and are equal.
A small disk centered near a homoclinic point includes infinitely many periodic points of different periods. Poincaré showed that if there is
a single homoclinic point, there are an infinite number. More specifically, there
are infinitely many homoclinic points in each small disk (Nusse and Yorke 1996).
