Increasing a slightly (to a=3.15 gives a clearer picture of the 2-cycle. Start from various initial conditions by clicking somewhere on the running plot, and occasionally hit "Clear" to remove old iterations.
Looking at the the second iterate of the map f^{ 2}(x) and starting from three different initial values shows that the fixed point near x=0.67 is unstable, and the orbit approaches one or other of the fixed points near x=0.55 and x=0.75.
x_{0}=0.85 | |
x_{0}=0.5 | |
x_{0}=0.69 | |
x_{0}=0.67 |
These two fixed points in f^{ 2}(x) are the two points x_{1}, x_{2} in the two cycle in f(x), since f(f(x_{1})=f(x_{2})=x_{1}.