Returning to the traditional value of b=0.3 we can follow the sequence of period doubling bifurcations. Here is the period 2 orbit.
It is interesting to see the underlying structure approximating the quadratic map by showing the transient behavior. The contraction onto the curve that will become the Henon attractor is more rapid than the attraction to the fixed points on this curve. So it we set the transient value to 3, and then click on the running plot to see the transient starting from various initial conditions, a portion of this curve is revealed.