Planar Maps

After about a dozen chapters on differential equations, we return here to the theme of Chapter 3 and explore, this time, some of the basic dynamics and bifurcations of planar maps. Our motives for delving into planar maps arc akin to the ones for studying scalar maps; namely, as numerical approximations of solutions of differential equations or as Poincaré maps. We begin our exposition with an introduction to the dynamics of linear planar maps. Then, following a section on linearization, we turn to numerical analysis and give examples of planar maps arising from “one-step” approximations of planar differential equations or from “two-step” approximations of scalar differential equations. Afterwards, we undertake, as usual, a detailed study of bifurcations of fixed points, including the Poincaré-Andronov-Hopf bifurcation for maps. The final part of the chapter is devoted to a synopsis of area-preserving maps, an important class arising from classical mechanics and possessing a rich history. The subject of planar maps is a vast one that is also mathematically rather sophisticated. Yet, many planar maps with innocuous appearances continue to defy satisfactory mathematical analysis. Indeed, the purpose of this modest, albeit long, chapter is to acquaint you with several famous planar maps and encourage you to explore their dynamics on the computer; for further mathematical nourishment, we will refer you to other sources.