Henon map: theoretical points





Presentation

The Henon map is a prototypical 2-D invertible iterated map with chaotic solutions proposed by the French astronomer Michel Henon (M. Henon, Commun. Math. Phys. Phys. 50, 69-77 (1976)) as a simplified model of the Poincare map for the Lorenz model. This simplified model consists in transforming the equations continuous in time in a discrete iterated system. Here is the Henon map:

xn+1 = 1 + axn2 + yn
yn+1 = bxn

The Henon map can be written in terms of a single variable with two time delays:

xn+1 = 1 + axn2 + bxn-1

The parameter a controls non-linerarity and the parameter b is a measure of the rate of area contraction (dissipation): for each iteration, areas are multiplied by a factor |b|: it means that if |b| < 1, an area of value 1 at the iteration k becomes an area of value |b| at the following iteration k+1.
and the Henon map is the most general 2-D quadratic map with the property that the contraction is independent of x and y. For b = 0, the Henon map reduces to the quadratic map, which is conjugate to the logistic map. Bounded solutions exist for the Henon map over a range of a and b values, and a portion of this range (about 6%) yields chaotic solutions as shown below:

The usual values used to produce chaotic solutions are a = -1.4, b = 0.3: despite the fact that the initial equations are very simple,  numerical calculations have shown that for these values, the Henon attractor is a strange attractor, but it has never been mathematically proven.
 

Basin of attraction

The strange attractor with its basin of attraction in black is shown below; initial conditions in the white regions outside the basin are attracted to infinity.