## Henon map: theoretical points

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:x_{n}_{+1}=1 +_{ }ax_{n}^{2}+y_{n}y_{n}_{+1}=bx_{n}The Henon map can be written in terms of a single variable with two time delays:

x_{n}_{+1}= 1 +ax_{n}^{2}+bx_{n}_{-1}The parameter

acontrols non-linerarity and the parameterbis 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 iterationkbecomes an area of value|b|at the following iterationk+1.

and the Henon map is the most general 2-D quadratic map with the property that the contraction is independent ofxandy. Forb= 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 ofaandbvalues, 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.

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.