Homoclinic structures and horseshoe

Smale horseshoe

The grey square with vertices (± 3.9, ± 3.9) (we consider mapping in the square further) is mapped into the light blue horseshoe. After the second iteration we get the dark blue double horseshoe. In a similar way inverse mapping makes light green and then dark green double horseshoes. Points form the right half of the square are mapped into the top light blue strip ant points from the left half are mapped into the bottom strip (test that by mouse or look at colored horseshoe in the Henon map). After the second iterations two light blue strips from the right (left) half of the square are mapped in two dark blue strips within the top (bottom) ligt blue band.
Click mouse with <Alt>/<Ctrl> to zoom the image. Center of the small square is mapped into the point marked by the big cross.
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. They are painted in pink and red colors. After every new iteration in every region appear four new intersections (press the "+" button to see this). Fractal Cantor repeller will appea for N → ∞.

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. You can test by mouse that the stable separatrises are blue and unstable ones are red.
The small square ih the center is mapped into the region with inverse colors. Drag the square to see where interesting point is mapped. Click mouse with <Alt>/<Ctrl> to zoom the image.
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 , g3 ...
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.

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updated 5 July 07