Forced Damped Pendulum
...with Four Basins
Even something as simple as a periodically forced damped pendulum can have
complex behavior. The computer-generated images below show initial positions
that are attracted to the different basins (one color for each basin). For
example, orbits starting at points in the blue region would yield a different
type of periodic motion from orbits starting in the red region. The brighter
the shade of color, the longer it takes to settle into the corresponding motion.
The different regions are separated by
fractal basin boundaries. The pictures are
at various magnification levels with the latter pictures being more magnified.
...with Two Basins
Same as above except the parameter values are slightly different so there is
only two attractors in this case instead of four. Here, lighter shades of color
represent points that settle down more slowly.
There are three simple attractors (straight lines in fact!) which formed sort of a
triangle; however, the basins of attraction is anything but simple. These pictures
are the starting points for the interesting ideas of riddled basins.
While many chaotic systems possess boundaries separating
basins that are typically fractal, they may also possess the stronger
property that any point which is on the boundary of one basin
is also simultaneously on the boundary of all the other basins.
This interesting property is known as the Wada property.
Wada boundaries are to be contrasted with more usual boundaries such as
those on a map showing the countries of Europe: there is only a finite
number of boundary points that are on the boundary of more than two
countries. A simple system exhibiting this interesting Wada property
is the three-disk model studied extensively in the context of
For more information about Wada basin boundaries in this system,
check out the paper "Wada Basin Boundaries in Chaotic Scattering."
This is not exactly a basin picture, but I didn't feel like creating
a new page just for this one picture. However, this is still a special
picture for me as it is my first published chaos artwork! For details,
check out the paper "Controlling Complexity."