Genericity and Stability of Morse-Smale Vector Fields

As we have emphasized before, the central objective of the Theory of Dynamical Systems is the description of the orbit structures of the vector fields on a differentiable manifold. There exist, however, fields with extremely complicated orbit structures as the example in Section 3 of Chapter 2 shows. Thus the strategy this programme must adopt is to restrict the study to a subset of the space of vector fields. It is desirable that this subset should be open and dense (or as large as possible) and that its elements should be structurally stable with simple enough orbit structures for us to be able to classify them. As far as the local aspect is concerned this problem is completely solved as we saw in Chapter 2.