Am example of the computation of the Lyapunov exponent for a one-dimensional dyanmical system.

We first enter the function that defines the dyanamical system, and its derivative.

Clear[x]

f[x_] = 
      Which[x < (1/2), x + 0.5, x > (1/2), 2.0 - 2*x, True, 
        0]

df[x_] = 
      Which[x < (1/2), 1.0, x > (1/2), -2.0, True, 1]

Plot[f[x], {x, 0, 1}]

Plot[df[x], {x, 0, 1}]

Now enter the initial point on the orbit and the number of iterations, M.

x[0] = 0.263971111675

0.3

M = 50

Now compute the points in the orbit and the Lyapunov exponent.

Do[{x[j + 1] = f[x[j]]}, {j, 0, M}]

lyap = (1/M)*Sum[Log[Abs[df[x[j]]]], {j, 0, M - 1}]