Invertible 2D Henon map is
x' = a + x2 + 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 + x2 + y
reflection in the diagonal y = x
x' = y,
y' = x.
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 Ws
is the set of all points that approach to the point under iteration of the map.
Similarly, the unstable manifold Wu 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 Wu. 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 x1 are shown below.
Let stable and unstable separatrises intersect in a homoclinic point
go . This point lies on stable separatrix so its orbit
goes to the saddle. As since the orbit passes g1 =
f(go ) then g1 belongs ws
too. Under inverse iterations go orbit go to the saddle
along the unstable separatrix. As since g1 =
f-1(go ) then "inverse" g1 orbit
go to x1 and the point lies on unstable separatrix too.
So it is one more intersection of ws and wu .
Therefore there are infinite number of intersections g2 ,
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 go . 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 go . Therefore
f -o(k+m) maps A in B and makes horseshoe
as shown in the figure to the left.
Intersection points of ws and wu for
two different saddles are called heteroclinic. One can make the
horseshoe map and chaotic fractal set these points too.
Previous: Smale horseshoe
Next: Bifurcation diagram
updated 3 July 2007