In section 1.4.5 we saw how a measurement of finite
precision in the bouncing
ball system has little predictive value in the long term.
Such
behavior is typical of motion on a chaotic attractor.
We called such behavior sensitive
dependence on initial conditions. For the special value 
in the
quadratic map we can analyze this behavior in some detail.
Consider the
transformation 
applied to the quadratic map when
. Making use of the identity
we find
This last linear difference equation has the explicit solution
Sensitive dependence on initial conditions is easy to see in this example when we express the initial condition as a binary number ,
Now the action of equation (2.31) on an initial condition
is a shift map. At each iteration we multiply the previous iterate
by two (10 in binary), which is a left shift, and then apply the mod
function, which erases the integer part. For
example, if
in binary, then
and we see the precision of our initial measurement evaporating before our eyes.
We can even quantify the amount of sensitive dependence the system exhibits,
that is, the rate at which an initial error grows. Assuming that our initial
condition has some small error 
, the growth rate of the error is
If we think of n as time, then the previous equation is of the form
with 
.
In this example the error grows at a constant exponential rate of 
.
The exponential growth rate of an initial error is the defining
characteristic of motion on a chaotic attractor. This rate of growth is called
the Lyapunov exponent .
A strictly positive Lyapunov exponent, such as
we just
found, is an indicator of chaotic motion. The Lyapunov
exponent is never strictly positive for a stable periodic motion.
