To name a periodic orbit, we need only choose one of its cyclic permutations. The number of distinct periodic orbits grows rapidly with the length of the period. The symbolic names for all periodic orbits up to period eight are presented in Table 2.3.
Table 2.3: Symbolic names for all periodic orbits up to period eight occurring
in the quadratic map
for 
. All names related by a cyclic permutation
are equivalent.
A simple indicator of the complexity of a dynamical system is its topological entropy . In the one-dimensional setting, the topological entropy, which we denote by h, is a measure of the growth of the number of periodic cycles as a function of the symbol string length (period),
where 
is the number of distinct periodic orbits of length n.
For instance,
for the fully developed quadratic
map, is of order 
, so
The topological entropy is zero in the quadratic map for any value of 
below the accumulation point of the first period doubling cascade because
is of order 2n in this regime.
The topological entropy is a continuous, monotonically increasing function
between
these two parameter values. The topological entropy
increases as periodic orbits are born by different bifurcation
mechanisms. A strictly positive value for the topological entropy is sometimes
taken as an indicator for the amount of ``topological chaos.''
In addition to its theoretical importance, symbolic dynamics will also be
useful experimentally. It will help us to locate and organize the periodic
orbit structure arising in real experiments. In Chapter 5 we will show how
periodic orbits can be extracted and identified from experimental data. We will
further describe how to construct a periodic orbit's symbolic name directly from
experiments and how to compare this with the symbolic name found from a model,
such as the quadratic map. Reference [14] describes an
additional refinement of
symbolic dynamics called kneading theory, which is useful for
analyzing
nonhyperbolic parameter regions, such as occur in the quadratic map
for 
.
Notice that the ordering relation described by the alternating binary
tree between the periodic orbits does not change for any . A simple
observation, which will nevertheless be very important from an experimental
viewpoint, is the following: this ordering relation, which is easy to
calculate in the hyperbolic regime, is often maintained in the nonhyperbolic
regime. This is the case, for instance, in the quadratic map for all 
. This observation is useful experimentally because it will give us a way to
name and locate periodic orbits in an experimental system at parameter values
where a nonhyperbolic strange attractor exists. That is, we can name and
identify periodic orbits in a hyperbolic regime, where the system can be
analyzed analytically, and then carry over the symbolic name for the periodic
orbit from the hyperbolic regime to the nonhyperbolic regime, where the system
is more difficult to study rigorously. Symbolic dynamics and periodic orbits will
be our ``breach through which we may attempt to penetrate an area hitherto
deemed inaccessible'' [15].
The reader might notice that our symbolic description of the quadratic map used very little that was specific to this map. The same description holds, in fact, for any single-humped (unimodal) map of the interval. Indeed, the topological techniques we described here in terms of binary trees extend naturally to k symbols on k-ary trees when a map with many humps is encountered.
This concludes our introduction to the quadratic map. There are still many mysteries in this simple map that we have not yet begun to explore, such as the organization of the periodic window structure, but at least we can now continue our journey into nonlinear lands with words and pictures to describe what we might see [16].
