MATH 164, Spring 2001 Due Date: Name(s):
Extra Project 10.5d: Differential vs. Difference Equations
(Value: 10 pts)
Suppose you're interested in studying the differential equation
dx
dt
= 4x
3
4
- x , x(0) = x0  (0, 1) (1)
and you've forgotten that it is just the logistic equation. (See the Epilog below.) If the only tools at your
disposal are numerical (read MatLab here, as opposed to Maple) then you might be tempted to approximate
solutions to equation (1) by solutions to the corresponding difference equation
xn = xn+1 - xn = 4xn
3
4
- xn , x0  (0, 1) (2)
which can be obtained via the iterative scheme
xn+1 = 4xn
3
4
- xn + xn = 4xn(1 - xn), x0  (0, 1)
or
xn+1 = 4xn(1 - xn), x0  (0, 1). (3)
Unfortunately we're now faced with a dilemma: On one hand, we know that all solutions x = x(t) to
equation (1) approach 3/4 as t  . (See Section 10.5 of the text.) On the other hand, in MATH 163's
Project 0.6b we observed, in discussing equation (3), that, "for some values of x0  [0, 1], xn converges [as
n  ], for some it repeats, and for others it wanders aimlessly over the interval [0, 1]." exhibiting chaos!
Problem: Explain what's going on here.
Epilog: The same issues raised in this project might well arise with equations other than the logistic
equation. The reason we singled out the logistic equation in this project is that we know what its solutions
look like, so it's easy to pose the dilemma posed above.