The Density Theorem for Structurally Stable Systems on a Two Dimensional Manifold

Abstract

Publisher Summary This chapter presents the density theorem for structurally stable systems on a two dimensional manifold. The structurally stable vector fields on a differentiable two manifold M2 are the ones that exhibit the simplest features. These are classified into equivalence classes modulo homeomorphisms of M2 onto itself mapping trajectories onto trajectories. The density theorem shows that for dimension two, the structurally stable vector fields are generic. The proof depends on several approximation lemmas. The crucial fact being whenever there is a minimal set μ that is not a singular point or a closed orbit, then by means of a C1 small perturbation, a closed orbit or else a new connection between saddle points is used as exit points.