Mathematical Biology Help!!!!

Hi, i am really stuck with a problem on bifurcation so if any one could help it would be very much appreciated!

Basically i have the function x_t+1=ax_t(1-x_t^2)
and i need to predict the value of a for the first bifurcation to a period 2 cycle occurs.
a is between 0 and (3*3^(1/2))/2
the eigen values i have are x_t=1 then eigen=-2a which is stable if 0 <=a <1/2 which is the only value that can be periodic i think??

Can anyone help??