One-Parameter Families of Vector Fields on Two-Manifolds: Another Nondensity Theorem

Abstract

This chapter discusses one-parameter families of vector fields on two-manifolds. The structurally stable Cr vector fields, r > 1, on a compact two-dimensional manifold are characterized by the following properties: (1) the closed orbits and singular points are all elementary, (2) there are no orbits that connect two saddle points, (3) the α- and ω-limit sets of each orbit are either singular points or closed orbits. It presents certain nonstable vector fields without rejecting a generic context. The framework in which the nonstable vector fields are pursued is sometimes called bifurcation theory. A nonstable vector field X will be embedded in a parametrized family of vector fields. As the dimension of the family increases, more highly degenerate vector fields will occur within a generic set of families. The chapter also describes techniques and lemmas used to prove Peixoto's theorem.