Real iteration

Even though the fixed point equation for a quadratic function is one of the simplest that there is, there is already considerable ambiguity about how it should be written. In its most general form, a second degree polynomial can be written in the form

p(x)= ax²+bx+c,

with three parameters. However the invertible mapping

$$r(x)=\alpha x + \beta$$

can be composed with p(x) to produce any desired change of origin and scale, since

$$p(r(x))=(a\alpha^2)x^2+\alpha(2a\beta+b)x+\alpha\beta^2+b\beta+c.$$

Although $\alpha$ is effectively determined by a, $\beta$ can be used to remove the linear term by solving a linear equation, the constant term by solving a quadratic equation, or to fulfill some other requirement on the coefficients. Knowing the geometric effect of the substitution and realizing that f represents a parabola, it is not difficult to visualize the effect of the transformation. One of the three parameters remains to make distinctions which cannot be accounted for by such a simple change of axis.

Even so, different canonical forms of the second degree fixed point equation

f(x)=x

are convenient in their own contexts. If emphasis is placed on mapping the unit interval into itself, the form

$$f(x)=\mu x (1-x)$$

is preferred. It is readily deduced that zero is always one fixed point and that the other is

$$x=\frac{\mu - 1}{\mu}.$$

That x=1 is approached asymptotically from below as a fixed point is a consequence of the zero of f at x=1; for larger $\mu$ the parabola rises more and more steeply through this point.

Stability of the fixed point is the fundamental concern of the theory of iterations; according to whether the magnitude of the derivative of f is smaller than 1 or not, iteration in its vicinity will be stable. In the present case,

$$f'(x)=\mu(1-2x)$$

whose value at the fixed point is

$$f'(1-\frac{1}{\mu})=2-\mu.$$

Therefore the fixed point is stable only in the region $1\leq\mu\leq3$.

At one time the discussion would have finished with this observation; letting curiosity motivate some inquiry just above this interval will show that successive iterates may oscillate about the fixed point, settling towards consistent underestimates followed by consistent overestimates. Effectively, there is a range within which two stable fixed points of the iterated function

g(x)=f(f(x))

bracket the unstable fixed point of f itself.

According to the chain rule of differentiation,

$$g'(x)=f'(f(x))\times f'(x),$$

so it is quite possible that the absolute value of this quantity does not reach 1 slightly away from the unstable fixed point. Nevertheless it is likely to close to 1, passing 1 if $\mu$ increased. In that case the fixed point of the iterate would also lose its stability, but with the possibility that still newer fixed points of

h(x)=g(g(x))

would be stable. Such reasoning quickly inspires the conjecture that there might be an infinite sequence of fixed points, progressively destabilizing as $\mu$ increases while creating a new pair which promptly splits away from it.

Less evident, but still understandable, is the fact that it is possible to run through the whole infinite series of splittings within a finite range of the parameter mu, leaving open the interesting question of what kinds of transfinite behavior to expect beyond such a limit.

 

Figure: three iterates of f(x) for $\mu =3.1$

Figure 4 shows the first three iterates of f(x) for the value $\mu=3.1,$ which lies just beyond the value of $\mu$ for which the fixed point becomes unstable; it is clear that the second and higher iterates still have stable fixed points.

Examining the graph confirms two useful properties of iterates. First, the fixed points of any function are automatically fixed points of its iterates; clearly f(a)=a implies f(f(a))=a. Second, critical points of a function are also critical points of its iterates; by the chain rule if f'(a)=0 then (f(f(a))'=f'(f(a))f'(a)=0.

At the fixed points, derivatives of the iterates are powers of the derivative of the function, making the stability of their common fixed points consistent for all the iterates.

Further examination of Figure 4 suggests that the higher iterates become squarer and squarer, their flat portions coinciding with the two fixed points of the second iterate. The suggestion is confirmed both by calculating the higher iterates, and by increasing $\mu$ slightly, causing the flatness to appear with earlier iterates. Indeed, reducing $\mu$ ever so slightly would make the second iterate more closely tangent to the diagonal, indicating that the onset of the instability of the original fixed point, and the rapid separation of the two fixed points of the iterate, depend sensitively on the parameter $\mu.$

It would seem that a second graph, showing the location of the fixed points as a function of the parameter $\mu,$ would serve as a map giving an overall view of the process, playing the same role that the Mandelbrot set plays for complex iterations.

 

Figure: fixed points as a function of $\mu$

Figure 5 shows such a graph, which was obtained experimentally by calculating a very high iterate of f, and then forming a histogram of the values of successive iterates. All values which had a sufficiently high frequency were marked on the graph.

One of the striking features of Figure 5 is the fact that the proliferation of fixed points for higher iterations of f has reached an infinite limit at a finite value of $\mu,$ and that the graph has further structure on beyond such a limit. Much of the recent work on the theory of iterations has been devoted to understanding this region of the diagram.