From our definition of a period n point, namely,
we see that finding a period n orbit for the quadratic map
requires finding the zeros for a polynomial of order 
. For instance, the
period one orbits are given by the roots of
which is a polynomial of order 2. The period two orbits are found by evaluating
which is a polynomial of order 4. Similarly, the period three orbits are given by solving a polynomial of order 8, and so on. Unfortunately, except for small n, solving such high-order polynomials is beyond the means of both mortals and machines.
Furthermore, our definition for the stability of an orbit says that once we
find a point of a period n orbit, call it 
, we next
need to evaluate the derivative of our polynomial at that point.
For instance, the stability of a period one orbit is determined
by evaluating
Similarly, the stability of a period two orbit is determined from the equation
Again, these stability polynomials quickly become too cumbersome to analyze as n increases.
Any periodic orbit of period n will have n points in its orbit. We will
generally label this collection of points by the subscript
n-1,
so that
where i labels an individual point of the orbit.
The
boldface notation indicates that 
is an n-tuple of real numbers.
Another complication will arise: in some cases it is useful to write our
indexing subscript in some base other than ten. For instance,
it is useful to work in base two when studying one-humped maps.
In general, it is convenient to work in base n+1 where n is
the number of critical points of the map.
It will be
advantageous to label the orbits in the quadratic map according
to some binary
scheme.
Lastly,
the question arises:
which
element of the periodic orbit do we use in evaluating the stability of an orbit?
In Problem 2.13 we show that all periodic points in a periodic orbit
give the same value for the stability function, 
[3]. So we can use any point in the periodic sequence. This fact is good to
keep in mind when evaluating the stability of an orbit.
