Planar Autonomous Systems

With this chapter we commence our investigation of the geometry of planar autonomous differentia] equations. After pointing out how such equations arise in applications, we develop some necessary generalizations of certain geometric ideas which are reminiscent of the ones explored earlier for scalar equations. Because the simplest examples of planar systems are constructed by bundling a pair of scalar equations— product systems—we present a discussion of such systems, including the Flow Box Theorem. We also analyze the geometry of conservative systems as another class of vector fields with special properties. Finally, to give a hint of things to come, we present multiple examples of autonomous differential equations illustrating various bifurcations on the plane.