Period-doubling route to chaos for the Hénon Map

Using eigenvalues of the Jacobian, show that the Hénon map undergoes a bifurcation from period-one to period-two exactly when \alpha = (3(\beta-1)^2)/4.
You can check your work by clicking the buttons below.

The period-doubling route toward the Hénon attractor can be observed in the following animation for \beta =0.4.

As \alpha varies from 0.2 through 1.24, wee see period-doubling, a four-piece attractor, a period-10 sink, and a two-piece attractor.
The two pieces finally merge to form the one-piece attractor.
b=0.4 animation

Here is the animation for \beta=0.3:
b=0.3 animation

Click here for the area-contracting feature and the inverse of the Hénon map as well as the box-counting dimension of the Hénon attractor. endamath