Linearization and Global Dynamics

Abstract

In this paper we show how the spectral theory of linear skew-product flows may be used to study the following three questions in the qualitative theory of dynamical systems: (1) when is an -limit set or an attractor a manifold? (2) Under which conditions will a dynamical system undergo a Hopf-Landau bifurcation from a k-dimensional torus to a (k + 1)-dimensional torus? (3) When is a vector field i the vicinity of a compact invariant manifold smoothly conjugate to the linearized vector field and how smooth is the conjugacy?