A perioddoubling
bifurcation occurs when 
the graph of L(x) passes from inside the blue bowtie to outside,
crossing over the diagonal line. 
This introduces two new fixed points, both stable, at least before
s increases very much. 
For this 2cycle, the original
fixed point (a 1cycle) becomes unstable at the same svalue
that the 2cycle appears. This is the source of the name perioddoubling. 

Click the picture for an animation. 