A tangent bifurcation occurs when the
graph of L(x) crosses the diagonal as s increases. 
The crossings occur where the
graph becomes tangent to the diagonal, hence the name tangent bifurcation. 
As s continues to increase, the graph crosses the diagonal at two points, initially
nearby, but moving apart. 
All this happens because as s increases, the
maxs of the graph of L(x) increase and the mins decrease. 

For the graph of L^{3}(x), three
lobes become tangent to the diagonal
at the same svalue, so this tangent bifurcation gives rise to six new fixed
points of L^{3}(x), hence to two 3cycles of L(x). See the left
picture. 
Looking carefully at the
crossing, we see one of the new fixed points is stable (the one toward the extremum in
the lobe of the graph crossing there), the other is unstable. See the right
picture. 



Click the picture to animate. 

Click the animation to stop. 

The stable fixed points of
L^{3}(x) constitute one 3cycle (light blue below) of L(x), the unstable
fixed points another 3cycle (dark blue). 
