Click mouse with <Alt>/<Ctrl> to zoom the image. Center of the small square is mapped into the point marked by the big cross. |
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. 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. |
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. |