Hyperbolic Theory

Abstract

The chapter is an introduction to the theory of hyperbolic structures in differential equations. The basic idea might be called “the principle of hyperbolic linearization.” Namely, if the linearized flow of a differential equation has “no eigenvalues with zero real parts,” then the nonlinear flow behaves locally like the linear flow. This idea has far-reaching consequences that are the subject of many important and useful mathematical results. Here we will discuss two fundamental theorems: the center and stable manifold theorem for a rest point and Hartman’s theorem.