The Hénon Map
In this site, we will explore the Hénon map, a two-dimensional iterated map function given by:
x_(n+1)=1+y_(n)-\alpha x_(n)^2 and y_(n+1)=\beta x_(n) where \alpha >0 and |\beta| < 1.
In 1976, in his article, Numerical study of quadratic area-preserving mappings, Michel Hénon proposed the Hénon map as a simple model of the Poincaré map for the Lorenz system.
The most representative form of the Hénon map has \alpha=1.4 and \beta=0.3. This Hénon map has a chaotic attractor as you can see below:
Download the MATLAB m-file, Henon_map.m and generate other Hénon maps by submitting \alpha and \beta values!
Or, if you prefer a MATLAB GUI, download PlotHenon.m
Not all Hénon maps have chaotic attractors, of course.
The fixed points of the Hénon map can be found by solving the equations, x=1-\alpha x^2+y and y=\beta x.
By substituting y=\beta x into x=1-\alpha x^2+y and using the quadratic formula, we find the fixed points to have coordinates:
x=(\beta-1+-sqrt((1-\beta)^2+4\alpha)) / (2\alpha) and y=\beta((\beta-1+-sqrt((1-\beta)^2+4\alpha)) / (2\alpha))
Thus, the Hénon map has two fixed points if and only if (1-\beta)^2+4\alpha > 0.
For example, consider the Hénon map with \alpha=0.1875 and \beta=0.5.
There are two fixed points, A(-4,-2) and B(4/3,2/3).
The Jacobian is given by J=((-2\alpha x, 1),(\beta,0)).
The eigenvalues for A are \lambda_(1)\approx-0.28 and \lambda_(2) \approx 1.78, so A is a saddle point.
The eigenvalues for B are \lambda_(1)=-1 and \lambda_(2) =0.5, so B is nonhyperbolic.
Here is the graph that depicts this behavior:
Click here for more periodic behavior of the Hénon map.