Differential Equations: A Dynamical Systems Approach 18

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Traditional courses on differential equations focus on techniques leading to solutions. Unfortunately, most differential equations do not admit solutions that can be written in elementary terms, particularly nonlinear equations in IR" for n>1. About 100 years ago, Poincaré, in his seminal paper on the three-body problem, changed the face of differential equations. He took the view that a differential equation defines a family of parametric curves in IR"; the object of the theory is to understand the geometry and behavior of these curves. The first part of our own series focuses on differential equations in dimension one, where the possible complication of the geometry is fairly limited. Here we attempt to extend those methods to higher dimensions. The problems are much more difficult than in dimension one, and some chapters of the present volume are of a more advanced nature than in Part I, although most should be accessible to undergraduates. Poincaré's insights were so novel and deep that it took a very long time for even the professional mathematical community to catch up. Before computer graphics, no one would have suggested that this material could be the topic of an undergraduate course. We are now able to bring Poincaré's approach to the undergraduate curriculum: the possibility of exploring the material experimentally has changed the nature of the challenge. The companion software MacMath, and Extensions of MacMath, is designed to bring these notions to life.