First, a warm "Hello to all!"...and excuse my English, pls

I need very much to calculate the following limit (it's a Lyapunov exponent):

After some numerical estimations, it seems to me that L is a periodic function for r<10 and a linear function for r>10. In both casese I obtained L < r*ln(2).

But I need very much something more exact...

Thank you all!