A common technique used to measure sensitive dependence on initial conditions is considered next. Qualitatively, an iterator is said to be sensitive to initial conditions if the orbits that result from two initial conditions, which are arbitrarily close, are distinctly different. The technique most often used to detect this type of behavior involves computing the Lyapunov exponent of system (7). Over the real domain, the Lyapunov exponent of system (7) is defined as
The exponent of equation (8) represents the mean
exponential rate of divergence or contraction between
two nearby orbits.
Since 
is difficult to calculate for most iterators,
the Lyapunov exponent is usually expressed as
where 
and 
.
A positive Lyapunov exponent indicates error growth, which means that
the iterator being measured is sensitive to initial conditions. A zero
or negative Lyapunov exponent indicates either no dependence on initial
conditions, or a contractive iterator where small errors are damped with
each successive iteration.
