Singularities of Gradient Vector Fields and Moduli

Abstract

This chapter focuses on singularities of gradient vector fields and moduli. The chapter describes gradient vector fields—that is, vector fields X for which there exist a Riemannian metric g and a function V such that g ( X , – t ) = dV . In this study, the considerations are mainly loca1, so it is assumed that all these objects to be defined on R n . It is assumed that X has a singularity in the origin, so dV (0) = 0. It is known that generic gradient vector fields on compact manifolds are structurally stable. This is also true if the vector field depends on one parameter, while in low dimensions the result even remains valid with more parameters. However, there is an example to show that generic k -parameter families of gradients need not be structurally stable if k ≥8. The example is based on a configuration of two saddles with two orbits of nontransverse intersection of stable and unstable manifolds and on the equality of certain eigenvalues at the saddles. This configuration leads to a so called modulus of stability. The chapter explains modulus of stability. An example is presented, which suggests that all complications, known to exist for diffeomorphisms, can be expected to show up when studying isolated singularities of gradient vector fields.