We define fn(x) for a given function f(x) and a given starting value x0 by
By the chain rule we then have that
By taking logarithms and dividing by n on both sides we get
This convergence can be proved by using a particular mathematical technique. The limit is called the Lyapunov exponent of the dynamical system.
Now suppose that we have two (close) starting values x0 and y0. By using the mean value theorem we see that
The conclusion is that
a) if we have that
that is, the system is not sensitive to the starting value.
b) if then
diverges to infinity with exponential growth.