Introduction: Background for Ordinary Differential Equations and Dynamical Systems

Abstract

The purpose of this first chapter is to review and develop the necessary concepts from the theory of ordinary differential equations and dynamical systems which we will need for the remainder of the book. We will begin with some results from classical ordinary differential equations theory such as existence and uniqueness of solutions, dependence of solutions on initial conditions and parameters, and various concepts of stability. We will then discuss more modern ideas such as genericity, structural stability, bifurcations, and Poincaré maps. Standard references for the theory of ordinary differential equations are Coddington and Levinson [1955], Hale [1980], and Hartman [1964]. We will take a more global, geometric point of view of the theory; some references which share this viewpoint are Arnold [1973], Guckenheimer and Holmes [1983], Hirsch and Smale [1974], and Palis and deMelo [1982].KeywordsPhase SpaceOrdinary Differential EquationVector FieldPeriodic OrbitTangent SpaceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.