Periodic Behavior of the Hénon Map

Say, we fix \beta at 0.4.
We see fixed points for \alpha=0.2:
period 1
We see a period-two sink for \alpha=0.5:
period 2
We see a period-four sink for \alpha=0.9:
period 4

Since some orbits are unbounded and go off to infinity, the choice of the seed is important in visualizing the behavior of the Hénon map.
Henon_map.m takes (0.1,0) as its seed.
The bifurcation diagram plotted below illustrates the period-doubling route to the chaotic attractor.
bifurcation diagram for the Hénon Map

The MATLAB m-file, bifur_Henonmap.m creates the bifurcation diagram with the \alpha and \beta values as input.

If you prefer a MATLAB GUI, download Bifurcation_Henon.m and Bifurcation_Henon.fig

You may be wondering for what values of \alpha the period doubling bifurcations occur?
The next page samples an analytical proof of the first period-doubling bifurcation. endamath