Periodic Behavior of the Hénon Map
Say, we fix \beta at 0.4.
We see fixed points for \alpha=0.2:
We see a period-two sink for \alpha=0.5:
We see a period-four sink for \alpha=0.9:
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.
The MATLAB m-file, bifur_Henonmap.m creates the bifurcation diagram with the \alpha and \beta values as input.
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.