For the remainder of this chapter, we will consider the simplest nonlinear polynomials, the quadratics. Quadratic polynomials apparently have three parameters
Two of these parameters are illusory from the point of view of
dynamics. This comes about from the notion of conjugacy.
Take a linear map L(x)= ax+b with 
. Then L(x) is
one-to-one and onto, and has an inverse map 
. The function
is also a quadratic polynomial, with the added property that
for all m. Thus, a description of the orbits of g would be equivalent to a description of the orbits of f. We may choose a conjugating linear map L to simplify f.
One standard choice of quadratic maps is
This one is more commonly used in the discussion of complex quadratic polynomials. Another standard choice is
There is a relation between c and 
under which these two
quadratic polynomials are linearly conjugate. We leave this to the
reader for a rainy day's algebra amusement.
In this chapter, we will suppose all our quadratics have the form
This polynomial passes through the x-axis at x=0,1. Instead of
working with the whole real line, we would like to work entirely
within the unit interval I=[0,1]. If 
, then 
maps I onto negative numbers; therefore, we shall assume
. The maximum of the parabola occurs at 
,
with 
. Then 
. In order for the iterates to be confined to
I, we need 
or 
. In that case, we
call an interval self-mapping.
For small , the maximum is also small and the entire parabola
lies below y=x. Then the web diagram looks like this
figure. This happens precisely when
the tangent line at x=0 has slope less than 1, i.e. when
. In that case, x=0 is an attractive
fixed point, and all starting points in I lead to 0 under
iteration.
When 
, the curve initially exceeds y=x and then falls
below it. A second point of intersection between 
and
y=x occurs at 
. The web diagram for
and 
is shown in this
figure.
Figure 65: An attractive fixed point at 
.
The derivative of at the new fixed point is
Thus, 
for 
, while
for 
. 
is an
attractive fixed point for 
, but in the range 
we see spiral approaches to the fixed point, as in
this figure.
Figure 66: Spiral approach to the fixed point for .
At this point, we are at the brink of a waterfall. As we turn our knob
beyond 3, we begin to set a cascade in motion. Both fixed
points x=0 and are repelling. But could there
be another type of long term behavior? Trepidaciously, let's turn the
knob up to 
and see the result here. The web starts to approach the fixed
point, but then it is pushed back out until it settles into a square
orbit about the fixed point. What that means is that there are two
points 
and 
such that 
and 
.
Figure 67: At a cycle of period two emerges.
The second curve is the graph of 
. It intersects y=x
at its fixed points, two of which form the stable 2-cycle.
These points are examples of periodic points of period 2.
In general, a periodic point of period n is a point p
such that 
, that is, p is a fixed point of the n-th
iterate 
. The forward orbit of p cycles forever through
. The finite sequence
is called an n-cycle for f.
We say p is an attractive periodic point (or
stable) of F if p is an attractive fixed point of .
Observe that if , then 
for all 
. To
specify the period, we say p has prime period n if n is
the smallest number such that .
For just above 3, the fixed point of ``splits''
into an attractive 2-cycle 
. Let's solve precisely for
these values:
This is a quartic equation; the roots are the 2-periodic points and
the two fixed points 
. To find 
,
divide this polynomial by x and 
.
After thrashing with the algebra, we find
Setting this polynomial equal to 0, we obtain a quadratic equation whose two solutions are:
The stability of this 2-cycle depends on 
, j=1,2
(the value of the derivative is the same for all elements of the
cycle). By the chain rule
Solving the inequality 
for 
,
we find that
For in this interval, the forward orbit of any point in
I tends to the 2-cycle .
.