In the previous chapter graphs were used to study the periodic orbits of a mapping f : S → S in the special case that S is a subinterval of ℝ. In particular we showed how the points of intersection of the graphs of f and id give the period-1 points of f. In this chapter we use these graphs to study more complicated orbits.

We show how to represent the successive iterates of a point on a graph by introducing the idea of a cobweb diagram. Cobweb diagrams will be studied first for linear and affine mappings. This leads us to study orbits which converge to, or which diverge away from, a fixed point. Such orbits are said to be asymptotic to the fixed point.

We prove theorems on the dynamics of linear and affine mappings. Later we show how the dynamics of affine mappings can be used to approximate the dynamics of differentiable mappings near a fixed point.

COBWEB DIAGRAMS

Let f : S → S where S is a subinterval of the real numbers and let x0 ∈ S. Given the graph of f, how can we use it to produce the sequence of iterates x0, x1, x2, x3, … of x0 under f?

Well, the first iterate is easy: the usual construction of a typical point (x0, f(x0)) on the graph of f is illustrated in Figure 4.1.1.