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

with three parameters. However the invertible mapping

can be composed with

Although
is effectively determined by *a*,
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

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

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

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
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,

whose value at the fixed point is

Therefore the fixed point is stable only in the region .

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

bracket the unstable fixed point of

According to the chain rule of differentiation,

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 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

would be stable. Such reasoning quickly inspires the conjecture that there might be an infinite sequence of fixed points, progressively destabilizing as 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 4 shows the first three iterates of *f*(*x*) for the value
which lies just beyond the value of
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 slightly, causing the flatness to appear with earlier iterates. Indeed, reducing 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

It would seem that a second graph, showing the location of the fixed
points as a function of the parameter
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 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
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.