Geometric Theory of Dynamical Systems

This chapter establishes the concepts and basic facts needed for understanding later chapters.

In this chapter we shall analyse the local topological behaviour of the orbits of vector fields. We shall show that, for vector fields belonging to an open dense subset of the space X ^r (M), we can describe the behaviour of the trajectories in a neighbourhood of each point of the manifold. Moreover, the local structure of the orbits does not change for small perturbations of the field. A complete classification via topological conjugacy is then provided.

Let M be a compact manifold of dimension m and X ^r (M) the space of C ^r vector fields on M, r ≥ 1, with a C ^r norm. In Chapter 2 we showed that the set G₁ ⊂ X ^r (M), consisting of fields whose singularities are hyperbolic, is open and dense in X ^r (M). This is an example of a generic property, i.e. a property that is satisfied by almost all vector fields. In this chapter we shall analyse other generic properties in X ^r (M). The original proof of the results dealt with here can be found in [44], [82] and [107].

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.