October 28, 2004 --- Class 18 --- Energy Conservation; Logistic Map and Chaos
Activities:
Energy Conservation and the Euler Richardson Method
On the homework, you were asked to find the optimal value of dt for
solving a differential equation. To do this, you should plot the
change in energy for a number of dt values. I looked at values
from 10^-2 to 10^-8 using a double precision code. From 10^-2 to
10^-4, the error in energy drops like dt^3. For 10^-5, the curve
is not quite so smooth. For smaller values of dt, the improvement
is marginal (10-6), or energy conservation is not as good at with
dt=10^-5. Who wants to do 10 times as much work and get worse results?
With values of dt that are too small, the error comes from round off
error.
For a single precision code dt=10^-3 is already showing ragged
behavior, indicating that round off error is becoming significant.
With a time step of 10^-3, the double precision code is conserving
energy significantly better than the single precision code.
What Value of dt is Needed for a 0.1% Error in the Energy
You were asked to solve this problem for part of homework using two
values of the angular frequency. I showed a plot of the error in
the energy at the end of the first period for different values of
dt. We see how the error is proportional to dt^3. It is easy to
determine the constant of proportionality in each case. With this
knowledge, it is trivial to find the value of dt that results in an
error of 0.1%.
Chaos and the Logistic Map
We introduced the logistic map from
consideration of an insect population model.
P_{n+1} = P_n (A - B P_n)
Upon scaling the population, the population is
replaced by x which lies in the range [0,1]. The next generation
is given by
x <- 4 r x * (1-x) .
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.
In the next class, we shall see what the long term future of the
population is.