Take the definition apart and look at the pieces for
easier interpretation. For each k, tells
how much the function f is changing with respect to
its argument at the point
. Since
,
this derivative expresses the magnitude of change
in the transition from
to
. Taking the logarithm
changes the scale on which the orbit is being studied. Finally,
the limit of the average of the log of the derivatives over
n iterations is taken to provide a measure of how fast
the orbit changes as (discrete) time propagates.
To summarise, the Lyapunov exponent contains information
about the average rate of separation of neighbouring initial
points. A positive Lyapunov exponent is an indication of
chaotic behaviour.
Of course, the limit in question may not be easy to calculate.
In practise, it is often helpful to use a ``finite time''
approximation to get some idea of what the Lyapunov exponent
may be for a given orbit.
Explicitly, for an orbit orb(), the finite
time analog of the Lyapunov exponent
is