Properties
The existence of one homoclinic point implies the existence of infinite number of them. This comes from its definition: the interception of a stable and unstable set. Both sets are invariant by definition, which means that the forward iteration of the homoclinic point is both on the stable and unstable set. By iterating N times, the map approaches the equilibrium point by the stable set, but in every iteration it is on the unstable manifold too, which shows this property.
This property suggests that complicated dynamics arise by the existence of a homoclinic point. Indeed, Smale (1967) showed that these points leads to horseshoe map like dynamics, which is associated with chaos.
Read more about this topic: Homoclinic Orbit
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