# Hénon map

(Redirected from Hénon attractor)
Hénon attractor for a = 1.4 and b = 0.3
Hénon attractor for a = 1.4 and b = 0.3

The Hénon map is a discrete-time dynamical system. It is one of the most studied examples of dynamical systems that exhibit chaotic behavior. The Hénon map takes a point (xnyn) in the plane and maps it to a new point

${\displaystyle {\begin{cases}x_{n+1}=1-ax_{n}^{2}+y_{n}\\y_{n+1}=bx_{n}.\end{cases}}}$

The map depends on two parameters, a and b, which for the classical Hénon map have values of a = 1.4 and b = 0.3. For the classical values the Hénon map is chaotic. For other values of a and b the map may be chaotic, intermittent, or converge to a periodic orbit. An overview of the type of behavior of the map at different parameter values may be obtained from its orbit diagram.

The map was introduced by Michel Hénon as a simplified model of the Poincaré section of the Lorenz model. For the classical map, an initial point of the plane will either approach a set of points known as the Hénon strange attractor, or diverge to infinity. The Hénon attractor is a fractal, smooth in one direction and a Cantor set in another. Numerical estimates yield a correlation dimension of 1.25 ± 0.02[1] and a Hausdorff dimension of 1.261 ± 0.003[2] for the attractor of the classical map.

## Attractor

Orbit diagram for the Hénon map with b=0.3. Higher density (darker) indicates increased probability of the variable x acquiring that value for the given value of a. Notice the satellite regions of chaos and periodicity around a=1.075 -- these can arise depending upon initial conditions for x and y.

The Hénon map maps two points into themselves: these are the invariant points. For the classical values of a and b of the Hénon map, one of these points is on the attractor:

${\displaystyle x={\frac {{\sqrt {609}}-7}{28}}\approx 0.631354477,}$
${\displaystyle y={\frac {3\left({\sqrt {609}}-7\right)}{280}}\approx 0.189406343.}$

This point is unstable. Points close to this fixed point and along the slope 1.924 will approach the fixed point and points along the slope -0.156 will move away from the fixed point. These slopes arise from the linearizations of the stable manifold and unstable manifold of the fixed point. The unstable manifold of the fixed point in the attractor is contained in the strange attractor of the Hénon map.

The Hénon map does not have a strange attractor for all values of the parameters a and b. For example, by keeping b fixed at 0.3 the bifurcation diagram shows that for a = 1.25 the Hénon map has a stable periodic orbit as an attractor.

Cvitanović et al. have shown how the structure of the Hénon strange attractor can be understood in terms of unstable periodic orbits within the attractor.

Classical Hénon map (15 iterations). Sub-iterations calculated using three steps decomposition.

## Decomposition

The Hénon map may be decomposed into an area-preserving bend:

${\displaystyle (x_{1},y_{1})=(x,1-ax^{2}+y)\,}$,

a contraction in the x direction:

${\displaystyle (x_{2},y_{2})=(bx_{1},y_{1})\,}$,

and a reflection in the line y = x:

${\displaystyle (x_{3},y_{3})=(y_{2},x_{2})\,}$.

## One Dimensional Decomposition

The Hénon map may also be deconstructed into a one dimensional map, defined similarly to the Fibonacci Sequence.

${\displaystyle x_{n+1}=1-ax_{n}^{2}+bx_{n-1}}$

## Special Cases and Low Period Orbits

If one solves the One Dimensional Hénon Map for the special case:

${\displaystyle X=x_{n-1}=x_{n}=x_{n+1}}$

${\displaystyle X=1-aX^{2}+bX}$

Or

${\displaystyle 0=-aX^{2}+(b-1)X+1}$

${\displaystyle X={1-b\pm {\sqrt {b^{2}-2b+1-4a}} \over 2a}}$

In the special case b=-1, this is simplified to

${\displaystyle X={\pm {\sqrt {a}} \over a}}$

If, in addition, a is in the form ${\displaystyle {1 \over c^{n}}}$ the formula is further simplified to

${\displaystyle X=\pm c^{n/2}}$

In practice the starting point (X,X) will follow a 4-point loop in two dimensions passing through all quadrants.

${\displaystyle (X,X)=(X,-X)}$
${\displaystyle (X,-X)=(-X,-X)}$
${\displaystyle (-X,-X)=(-X,X)}$
${\displaystyle (-X,X)=(X,X)}$