# Henon map

Invertible 2D Henon map is

*x' = a + x*^{2} + by,

y' = x.

Inverse map is

*x = y',
*

y = (x' - y'^{ 2} - a)/b.

The map is decomposed in three simple operations:

uniform squeezing *b* times in the *y* direction
(to the left the squeezing is omitted, i.e. *b = 1*)

*x' = x,
*

y' = by,

bending in the same direction

*x' = x,
*

y' = a + x^{2} + y

reflection in the diagonal *y = x*

*x' = y,
*

y' = x.
## Smale horseshoe

You see below, how square region with vertices
*(± 3.9, ± 3.9)* is mapped into Smale horseshoe.
After the second iteration we get doubled horseshoe. In a similar
way inverse mapping makes veritcal horseshoe.
Points of the non-wondering set (which do not go to infinity and stay inside
the square forever) lie in intersection of two transversal horseshoes.
After every new iteration in every region appear four new
intersections. Fractal Cantor repeller
will appea for *N → ∞*.
Click mouse with <Alt>/<Ctrl> to zoom the image.
## Stable and unstable manifolds and homoclinic structures

For a hyperbilic fixed point of a map the stable manifold *W*_{s}
is the set of all points that approach to the point under iteration of the map.
Similarly, the unstable manifold *W*_{u} is the set of all orbits
that approach to the point under iteration of inverse map.
In 2D these are the saddle point and stable and unstable separatrises.
If we start with a small ball of initial points centered around a
saddle and iterate the map the ball will be stretched and squashed along
the line *W*_{u}. Similarly the small ball of initial points
iterated backward in time will trace the stable separatrises.
*N* iterations of a small circle (with radius *R*) around
the saddle *x*_{1} are shown below.

Let stable and unstable separatrises intersect in a *homoclinic* point
*g*_{o }. This point lies on stable separatrix so its orbit
goes to the saddle. As since the orbit passes *g*_{1} =
f(g_{o }) then *g*_{1} belongs *w*_{s }
too. Under inverse iterations *g*_{o} orbit go to the saddle
along the unstable separatrix. As since *g*_{1} =
f^{-1}(g_{o }) then "inverse" *g*_{1} orbit
go to *x*_{1 } and the point lies on unstable separatrix too.
So it is one more intersection of *w*_{s} and *w*_{u }.
Therefore there are infinite number of intersections *g*_{2 },
g_{3 }...
By increasing *N* you can test that separatrises are very complicated.
There are many intersection points and you can be entangled easy...

and it is an evidence of complex dynamics :)

To make it a bit severe we will show that there is Smale horseshoe
in a homoclinic structure.
Let a map has a homoclinic point *g*_{o }. We take a region
*D* around the saddle. *A = f*^{ ok}(D) will be stretched
along unstable separatrix and reach the homoclinic point at some
*k* value. Similar *B = f*^{ -om}(D) will be stretched
along the stable separatrix and reach *g*_{o }. Therefore
*f*^{ -o(k+m)} maps *A* in *B* and makes horseshoe
as shown in the figure to the left.
Intersection points of *w*_{s} and *w*_{u } for
two different saddles are called *heteroclinic*. One can make the
horseshoe map and chaotic fractal set these points too.

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*updated* 3 July 2007