ATTRACTOR SETS
In this page we describe the methods used to create the OUTPUT SETS when
The COMPOSITE button and the INDIVIDUAL button will do the same thing when only a single INPUT FUNCTION is selected. When multiple INPUT FUNCTIONS are selected the COMPOSITE button will follow the below algorithm B1 or 2. When multiple INPUT FUNCTIONS are selected the INDIVIDUAL button will follow the below algorithm A1 or 2 for each selected function individually. Thus if three INPUT FUNCTIONS are selected, then the INDIVIDUAL button will create three OUTPUT SETS, one for each selected INPUT FUNCTION.
1. SET TYPE = ATTRACTOR and PROCESS TYPE = RANDOM
A1. Single Function Iteration Algorithm (when only one INPUT FUNCTION is selected):
Note: Since only one INPUT FUNCTION is selected, there is actually no randomness involved. In this case, the Attractor set is simply a finite portion of the orbit of z0. Thus this set of options can be used to display the Iterates of a Seed Value by setting the number of SKIPS equal to 0.
I. Call
the selected INPUT FUNCTION by the name F(z)
the (complex number) SEED VALUE by the name z0
the number of POINTS TO PLOT by the name N
the number of SKIPS by the name S
II. Compute the complex numbers z1 = F(z0), z2 = F(z1), z3 = F(z2), ... Continue this process N+S times to get a sequence of complex numbers z0, z1,..., zN+S such that F(zn-1) = zn.
III. The set of points {zS+1, zS+2,..., zN+S}, formed by skipping the first S points of the sequence, is called the (Approximate) Attractor set of F(z). It is this set which is created as an OUTPUT SET and labeled with the map which generated it. For N and S large we get better approximations to the actual Attractor set of F(z).
B1. Multiple Function (IFS) Algorithm (when multiple INPUT FUNCTIONS are selected):
We follow the above algorithm, but with a slight change. Call the selected INPUT FUNCTIONS by the names G1,...,Gk and change item II above as follows. Starting with z0 we randomly choose a map F1 from G0, ..., Gk and then calculate z1 = F1(z0). Then randomly choose a map F2 from G1, ..., Gk and then calculate z2 = F2(z1). Iterate this process N+S times to get a sequence of complex numbers z0, z1, ..., zN+S. The set of points {zS+1, zS+2,..., zN+S}, formed by skipping the first S points of the sequence, is called the (Approximate) Attractor set of the IFS <G1,...,Gk> (acting in parallel). In the right window the OUTPUT SET is labeled the COMPOSITE ATTRACTOR SET.
2. SET TYPE = ATTRACTOR and PROCESS TYPE = FULL
This is the same algorithm whether one INPUT FUNCTION or multiple INPUT FUNCTIONS are selected:
I. Call
the selected INPUT FUNCTION by the name F(z)
the (complex number) SEED VALUE by the name z0
the number of POINTS TO PLOT by the name N
the number of SKIPS by the name S (this is not relevant for this process type)
II. Call the selected INPUT FUNCTIONS by the names G1,...,Gk. Calculate the set A1=G1({z0})
... ∪ ∪Gk({z0}). Then calculate the set A2=G1(A1) ... ∪ ∪Gk(A1). Continue this process until the set Aj=G1(Aj-1) ... ∪ ∪Gk(Aj-1) has at least N points.
Note that if only one INPUT FUNCTION was selected set, each of the sets Aj will only have one point. In this case the program is set to simply terminate with j=N. If multiple INPUT FUNCTIONS are selected, then the sets Aj will necessarily grow in size. Although this is not mathematically true (e.g., if each of the multiple input functions fixes the seed value), the computer will regard the repeated points calculated as distinct points in the set and therefore the size of the sets Aj will grow.The set of points Aj is called the (Approximate) Attractor set of the IFS <G1,...,Gk> (acting in parallel). In the right window the OUTPUT SET is labeled the COMPOSITE ATTRACTOR SET.