Introduction: Differential Equations and Dynamical Systems

Abstract

In this introductory chapter we review some basic topics in the theory of ordinary differential equations from the viewpoint of the global geometrical approach which we develop in this book. After recalling the basic existence and uniqueness theorems, we consider the linear, homogeneous, constant coefficient system and then introduce nonlinear and time-dependent systems and concepts such as the Poincaré map and structural stability. We then review some of the better-known results on two-dimensional autonomous systems and end with a statement and sketch of the proof of Peixoto’s theorem, an important result which summarizes much of our knowledge of two-dimensional flows.