Scalar Maps

With the current proliferation of computers, numerical simulations have become common practice, suggesting new mathematical discoveries and new areas of applications. Despite the power of numerical approximation schemes as “experimental” tools and their case of implementation on the computer, there is always the difficulty of deciding on the accuracy of computations. Even in the case of a scalar differential equation, one can he confronted with rather strange mathematical phenomena. This is largely due to the fact that numerical approximation of a differential equation leads to a difference equation, and that difference equations, despite their innocuous appearance, can haw amazingly complicated dynamics. In this chapter, we first illustrate how difference equations, also called maps, arise in numerical approximations. Because of their importance in other contexts, we then undertake the study of dynamics and bifurcations of maps. In particular, we investigate local bifurcations of a class of maps, monotone maps, which will later play a prominent role in our study of differential equations. We end the chapter with a brief exposition of a landmark quadratic map, the logistic map.