To really understand a dynamical system we should identify all
the attractors in its phase space. Then for each attractor we
should find all the possible combinations of initial conditions
which settle to that attractor. The set of points in the
(x,x') plane which as initial conditions lead to a particular
attractor are called that attractor's basin of attraction. A
section of the (x,x') plane with the attractor basins marked,
is called an "attractor basin (AB) map" of the system.
Since each point in the (x,x') plane goes to some attractor,
the map fills the entire plane.
The business of making an AB map for a dynamical system is
difficult. The only approach for a personal computer is to sample
a number of points in the (x,x') plane, using each as the
initial conditions for the system under study. Then to let the
system develop over time until it settles to an attractor. When
that happens we could make a record of the attractor found for
reference in subsequent samples, then color the pixel
representing the starting point in the (x,x') plane a
different color for each attractor. This takes too long to be of
much use. A 100 MHZ pentium computer might work for several days
on a high resolution AB map.
What we provide here is a quick way to identify which
attractor "owns" any specific point in the (x,x')
plane. Let's go back to the DMO
Periodic Attractor 3 display and select the Basins of
Next we will get into a different kind of attractor, neither
point nor periodic.
Are there any questions?