In the previous section we saw that for 
an infinite
number of periodic orbits with
period 
are born in the quadratic map. In a period
doubling cascade we know the sequence in which
these periodic orbits are born.
A period one orbit is born first, followed by a period two
orbit, a period four orbit, a period eight orbit, and so on. For higher
values of 
, additional periodic orbits come into existence. For
instance, a period three orbit is born when 
, as we showed in section 2.7.1. In this section, we will
explicitly show that all possible
periodic orbits exist for 
. One of the goals of
bifurcation theory is to understand the different mechanisms for the birth and
death of these periodic orbits. Pinning down all the details of an
individual problem is usually very difficult, often impossible. However,
there is one qualitative result due to Sarkovskii of great beauty that applies
to any continuous mapping of the real line to itself.
The positive integers are usually listed in increasing
order 
However, let us consider an alternative enumeration
that reflects the order in which a sequence of period n
orbits is created.
For instance, we might list the sequence of
integers of the form as
where the symbol 
means ``implies.''
In the quadratic map system
this ordering says
that the existence of a period orbit implies
the existence of all periodic orbits of period 
for i ;SPMlt; n.
We saw this ordering in the period doubling cascade.
A period eight orbit thus implies the
existence of both period four and period two orbits.
This ordering diagram says nothing about
the stability of any of these orbits,
nor does it tell us how many periodic orbits
there are of any given period.
Consider the ordering of all the integers given by
with 
.
Sarkovskii's theorem says that the ordering found in
equation (2.29) holds,
in the sense of the ordering above,
for any continuous map of the real
line R to itself--the existence of a period i orbit
implies the existence of all periodic orbits of
period j where j follows i in the ordering.
Sarkovskii's theorem is remarkable for its lack of
hypotheses
(it assumes only that f is continuous).
It is of great help in understanding the structure of
one-dimensional maps.
In particular, this ordering
holds for the quadratic map.
For instance, the existence of a period seven orbit implies the
existence of all periodic orbits except a period five and a period three
orbit. And the existence of a single period three orbit implies the existence
of periodic orbits of all possible periods for the one-dimensional map.
Sarkovskii's theorem
forces the existence of period doubling cascades in one-dimensional maps. It is
also the basis of the famous statement of Li and Yorke that
``period three implies chaos,'' where
chaos loosely means the existence of all possible periodic orbits.
An elementary proof of
Sarkovskii's theorem, as well as a fuller mathematical treatment of maps as
dynamical systems, is given by
Devaney in his book An Introduction to Chaotic
Dynamical Systems [10].
Sarkovskii's theorem holds only for mappings of the real line to itself. It does
not hold in the bouncing ball system because it is a map in two dimensions. It does
not hold for mappings of the
circle, 
, to itself. Still, Sarkovskii's theorem is a lovely result,
and it does point the way to what might be called ``qualitative universality,''
that is, general statements, usually topological in
nature, that are expected to hold for a large class of dynamical systems.
