We define *f _{n}(x)* for a given function

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 *x _{0}* and

since

and

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.