Qualitative approach to autonomous equations. (Next Section)
Bifurcations and Phase Lines (Cover Page)
Bifurcations and Phase Lines (Previous Section)
With the implementation of the NCTM Standards in many high schools across the country, many students are now entering college level mathematics courses with significantly different backgrounds and expectations. As dictated by the Standards, students are quite familiar with technology (especially graphing calculators). They are used to visualizing functions using their graphs, not just via their analytic formulas. And they have had significantly more exposure to varying parameters and witnessing qualitative changes in graphs.
All of this gives college mathematics instructors an excellent opportunity to revamp the existing differential equations curriculum so as to include more modern and relevant material, and to present traditional topics from a new viewpoint.
One such topic is elementary bifurcation theory. This topic is rarely included in traditional differential equations courses, yet it is of crucial importance in many engineering applications. In the Boston University Differential Equations project (a joint effort with Paul Blanchard and G.R. Hall), bifurcations form one strand which continually reappears throughout the course. Bifurcations initially arise in the study of first order autonomous equations. They appear again when the qualitative theory of linear systems is discussed. And they show up later in the course when nonlinear systems and discrete dynamical systems are treated. In this note, we describe how we teach the first of these topics.