Background from the Theory of Differentiable Manifolds

Abstract

Before discussing the general theory of invariant manifolds, we need to give some background material from differential geometry. More specifically, we will need to understand the definition of a differentiable manifold, the tangent space at a point, the tangent bundle, and the derivatives of maps defined on differentiable manifolds. These notions will be crucial for discussing the dynamics near an invariant manifold. We will not develop these concepts in the most abstract or mathematically crisp manner, but rather along the lines where they occur frequently in applications. In applications involving the modeling of the dynamics of some physical system, we typically choose certain quantities describing various aspects of the system and write down equations describing the time evolution of these quantities. These quantities constitute the phase space of the system with invariant manifolds often arising as surfaces in the phase space. Consequently, we choose to develop the concept of a differentiable manifold as a surface embedded in ℝ n (loosely following the exposition of Milnor [1965] and Guillemin and Pollack [1974]) and refer the reader to any differential geometry textbook for the abstract development of the theory of differentiable manifolds (e.g., a standard and very thorough textbook is Spivak [1979]). Our approach will allow us to bypass certain set-theoretic and topological technicalities since our manifolds will inherit much structure from ℝ n , whose topology is relatively familiar. Additionally, it is hoped that this approach will appeal to the intuition of the reader who has little or no experience with the subject of differential geometry.