All Planar Things Considered

Abstract

In the numerous chapters that have come before, we have encountered many bifurcations, such as saddle-node for equilibria and periodic orbits, Poincaré-Andronov-Hopf, and breaking homoclinic loops and saddle connections. It is natural to ponder when, if ever, we will stop adding to the list and produce a complete catalog of all possible bifurcations. In this chapter, we indeed provide such a list for “generic” bifurcations of planar vector fields depending on one parameter. However, due to the overwhelming difficulty of the subject matter, our exposition, while precise, is devoid of verifications. To circumvent, certain technical complications, we confine our attention to a closed and bounded region of the plane, and in such a region characterize the structurally stable vector fields. To motivate this confinement, we then make a short digression to describe a class of vector fields whose dynamics are naturally confined to a bounded region—dissipative systems. Next, we explore the geometry of sets of mildly structurally unstable vector fields—first-order structural instability. By determining the sets of such vector fields forming hypersurfaces in the set of all vector fields, we arrive at a list of one-parameter “generic” bifurcations. You will undoubtedly notice that some of the familiar bifurcations are absent from the list. We provide an explanation for this as well, in terms of symmetries. We end the chapter with a glimpse into the intricate bifurcations of two-parameter vector fields.KeywordsVector FieldPeriodic OrbitEquilibrium PointGlobal AttractorDissipative SystemThese 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.