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.