October 16, 2013---Class 15--- Chaos and Logistic Map, Finding Bifurcation Points
Activities:
Chaos and Logistic Map Continued
We continued looking at our population model, the logistic map.
A program entitled map allows you to follow the time history of a
population. Map takes three command line arguments: r, the
parameter that appears above in the definition of the map; x_0, the
initial value of the population; and iterations, the total number
of generations to calculate.
We have seen what happens after a long time if we use r=0.1.
The population dies out if we start with x=0.1
map 0.1 0.1 100
Students were asked to try different starting values. We then
constructed a shell script to loop over different starting values.
#!/bin/csh
set r = 0.1
foreach x (0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9)
map $r $x 100 >>! temp
end
When r is sufficiently small, no matter what the starting
population, after a long time the population approaches 0.
With r a little greater than 0.25, after many generations the
population approaches a nonzero constant for any starting value
other than 0 or 1.
With larger r values, we see that the final value is increasing.
With r=0.78, we find something very curious. The
population oscillates in time between two different values
5.475744e-01 and 7.729384e-01. You should verify that this does
not depend on the starting value (as long as it is not 0 or 1).
This result is called having an attractor of period 2.
Now, we are getting the idea that there can be different long time
behaviors. For small r, we go to zero; for larger r, we approach a
non-zero constant; for yet larger r, we oscillate between two
values.
With r=0.863, we found that the system has a periodicity of 4.
We would like to explore the long time behavior by looking at the
set of x values on each trajectory. To do this, we want to print
out a certain number of values on the trajectory starting after
a number of generations. A program called fixed_pts.c can do this.
It will do this for a range of r values. It arguments are
rmin rmax dr ntransient nprint
rmin is the minimum value of r to be considered, rmax is the maximum
and dr is the increment in r. ntransient is the number of initial
generations before we start printing x_n. nprint is the number of
generations to print after the transient generations.
With this program we were able to produce pictures of period doubling
and chaos as in Figure 6.2 on page 145.
We would like to better understand how period doubling comes about.
We are looking at the mapping
x_{n+1} = f(x_n)
and when r <= 0.75 we find that the long term behavior is a
constant. We call this a fixed point. Let
x_{n} = x_f + \epsilon_n
where x_f is the fixed point and \epsilon_n is how far we are from
the fixed point at the n th generation.
By Taylor expanding around x_f, we found that if |f'(x_f)| < 1, we
have a stable fixed point, otherwise the fixed point is unstable.
It was easy to find the fixed points from the equation
x_f = 4 r x_f (1-x_f)
There are two solutions, x_f = 0 and x_f = 1- 1/(4r)
We easily calculated the derivatives at the fixed points. For r <
0.25, x_f=0 is a stable fixed point, and the other value is not
between 0 and 1. For 0.25 < r < 0.75, x_f = 1- 1/(4r) is a stable
fixed point and x_f = 0 is unstable. When r > 0.75 neither is a
stable fixed point. However, when we have a trajectory of period
2, then x_(n+2) = f(f(x_)) can have a stable fixed point. We
explored this using Mathematica. Look in
~sg/Documents/February28_chaos.nb
to see what we did in Mathematica.
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 way 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.
Can you improve on this table?