With splitting you mean what happens directly after mu \approx 0.75?

I am plotting the orbit. So denote the logistic map by f_\mu. And what I plot is the set { f^n(x_0) : n \in N, n> big number } for x_0=1/2. So after \mu \approx 0.75 the orbit is an attracting period-2 orbit.

I am not sure if that answers your question :3