3. The Lyapunov exponent.

We define f_n(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

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.