October 25, 2007 --- Class 17 --- Finding Bifurcation Points
Finding Bifurcation Points
We are interested in accurately finding the bifurcation points
where we go from stable attractors of period 2^n to stable
attractors of period 2^(n+1). This can be done using the code
in fixed_pts.c in my home directory, which is compiled and in my
bin directory. Recall that we used this code to prepare a
diagram like Fig. 6.2 on page 134 of CSM. We also know that at
the bifurcation points, the convergence is very slow. We can
use this to iteratively run the code with narrower windows and
more initial iterations discarded to find the bifurcation
points.
For instance, if you run:
fixed_pts .7 .999 0.001 1000 128 |axis m 0 |xplot
you will get a nice diagram like Fig 6.2. If you look at the
output of fixed_pts, you will see that for each value of r,
there are 128 lines with the x values that result after 1000
iterations are discarded. Away from the r values that
correspond to bifurcation points, convergence is fast and many
of the x values are repeated as we loop over the points in
stable attractors of period smaller than 128. However, near the
bifurcation points, convergence can be slow and we will find
more than the expected number of unique x values. How can we
use this to find the bifurcation points?
Unix has a utility called sort that is very useful. The option
-u, which stands for unique, will throw out repeated lines. For
example, in the output of
fixed_pts .7 .999 0.001 1000 128 | sort -u | more
you will find output like this:
7.430000e-01 6.635262e-01
7.440000e-01 6.639785e-01
7.450000e-01 6.644295e-01
7.460000e-01 6.648794e-01
7.470000e-01 6.653278e-01
7.470000e-01 6.653279e-01
7.470000e-01 6.653280e-01
7.470000e-01 6.653281e-01
7.470000e-01 6.653282e-01
7.480000e-01 6.657665e-01
7.480000e-01 6.657666e-01
7.480000e-01 6.657667e-01
7.480000e-01 6.657669e-01
7.480000e-01 6.657670e-01
7.480000e-01 6.657671e-01
7.480000e-01 6.657673e-01
7.480000e-01 6.657674e-01
7.480000e-01 6.657675e-01
Notice that r= 0.743 to 0.746 only appear once. For those
values of r, convergence is complete (to the printed accuracy)
after 1000 iterations. However, starting with .747, converence
is so slow that we do not just have 1 or 2 values corresponding
to a stable attractor of period 1 or 2.
We can use awk to count the number of lines for each value of r.
Here is a little awk script pts.a from my home directory:
BEGIN {old_r=0; count=0;}
{if($1 != old_r) {print old_r,count; old_r=$1; count=1;} else ++count;}
END {print old_r,count;}
Combine the commands:
fixed_pts .7 .999 0.001 1000 128 | sort -u | awk -f pts.a |mo
Part of the output now looks like this:
7.440000e-01 1
7.450000e-01 1
7.460000e-01 1
7.470000e-01 5
7.480000e-01 106
7.490000e-01 128
7.500000e-01 128
7.510000e-01 41
7.520000e-01 2
7.530000e-01 2
7.540000e-01 2
7.550000e-01 2
7.560000e-01 2
and another part looks like this:
8.590000e-01 2
8.600000e-01 2
8.610000e-01 2
8.620000e-01 128
8.630000e-01 5
8.640000e-01 4
8.650000e-01 4
8.660000e-01 4
In this we we can find ranges of r that correspond to the
transitions between stable attractors of different periods. You
may then bracket these regions, decrease the granularity in r
and increase the number of initial iterations that are thrown
away. It is important to note that the first value of r for
which the number of unique values increases will likely be a
little below the bifurcaton point.
This process can be tedious. In the text, you will find a table
of the bifurcation points to 6 digit accuracy: Table 6.1 page 153.
Superstable trajectories
In our discussion of fixed point iteration, we found that if
x_{n+1} = f(x_n)
the rate of convergence near a fixed point is dependent on the
derivative of f at the fixed point. In our case, we are interested
not only in f(x)=4r x (1-x), but in the iterated functions f^(n)(x).
Because of the symmetry of f(x) around x=1/2, the iterated functions
all have a derivative of zero at x=1/2. When one of the fixed
points is at x=1/2, we have what we call a superstable trajectory.
We used Mathematica to find the values of r that lead to
to superstable trajectories, i.e., the trajectories that include
x=0.5.
Solve[ f[1/2, r] == 1/2, r]
will solve for the trajectory of period 1. You can define the
iterated functions, f2[x_,r_] := f[f[x,r],r] and
f4[x_,r_] := f2[f2[x,r],r], etc. and similarly solve for values of
r.
I discussed the program findroot.c that uses a routine from
Numerical Recipes to solve these equations numerically.
The routine is called zbrent and may be found in
Chapter 9 on Root Finding. This may be used on the homework on
problem 6.6. Quadratic interpolation is
used in the Brent algorithm, but is used to interpolate x as a function
of y. The desired x value is the one for which y=0. This results in
a complicated, but linear function, for x the next guess at the root.
The zbrent routine is useful when you can bracket the root and only
want to evaluate the function whose root you want, not its derivative.
You can read about this routine by looking in the file ~sg/ch9-3.pdf.
We downloaded this file from a web site at Cornell that has the text
of Numerical Recipes. You may easily find this site using Google.
Running my code, I find values of s_k different from those appearing
in the book in Table 6.2. My values are correct as you should be
able to verify by iterating the fixed point equation starting with
x=0.5 and see who gets closer to 0.5 after the appropriate number
of iterations.
I will look into converting this code to use root finding routines
from the Gnu Scientific Library.