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 isThe following examples and calculations are intended to illustrate these concepts more fully.