# The Hénon Map

amath 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!

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: