Heteroclinic tangle for a three dimensional volume preserving mapping.
Resonance (mode-locking) tongues are commonly observed in smooth circle maps, and Neimark-Sacker bifurcations.
The stunning term, "Blue Sky Catastrophe" has been given to the last, out of the seven known, main bifurcations of a periodic orbit.
The following pictures illustrate some elements of the nonlinear dynamics near equilibrium manifolds.