amath
## 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.

step endamath 1 : amath Recall that the Jacobian is J=((-2\alpha x, 1),(\beta,0)). The eigenvalues, \lambda_(1,2) , are given by det(J-\lambda I)=0.

step endamath 2 : amath The eigenvalues are \lambda_(1,2)=-\alpha x +- sqrt((\alpha)^2 x^2 +\beta).

step endamath 3 : amath A bifurcation occurs when |\lambda|=1. Take the case, \lambda=-1.

step endamath 4 : amath -1=-\alpha x +- sqrt((\alpha)^2 x^2 +\beta) simplifies to -2\alpha x +1=\beta

step endamath 5 : amath Substituting x=(\beta-1+-sqrt((1-\beta)^2+4\alpha)) / (2\alpha) and simplifying gives us \alpha = (3(\beta-1)^2)/4.

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.

Here is the animation for \beta=0.3:

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