MATH 261, Spring 2001 Due Date: Name(s):
Honors Project 10: Iterative Processes and Periodicity Revisited
In MATH 163, we briefly discussed iterative processes and periodicity in the context of single variable
calculus. Not only does much of the discussion of iterative processes extend to the context of several
variables, but the theory becomes considerably more interesting.
The following Maple code generates a graphic of the Lorenz system: this system is associated to the system
of first-order differential equations
dx
dt
= a(y - x)
dy
dt
= b(x - y - xz)
dz
dt
= xy - cz
where a = 10, b = 28, and c = 8/3 and x0 = 1, y0 = 1, and z0 = 10, or, equivalently, to the first-order
iterative process
xn+1 = a(yn - xn) + xn
yn+1 = b(xn - yn - xnzn) + yn
zn+1 = (xnyn - czn) + zn
for some small real number .
> # Honors Project 10: The Lorenz System
> restart: with(plots):
> a := 10; b := 28; c := 8/3;
> Delta := 0.01;
> f := (x,y,z) -> a*(y-x)*Delta+x;
> g := (x,y,z) -> (b*x-y-x*z)*Delta+y;
> h := (x,y,z) -> (x*y-c*z)*Delta+z;
> xold := 1.0; yold := 1.0; zold := 10.0;
> mylist := [xold,yold,zold]:
> for N from 1 to 4000 do
xnew := f(xold,yold,zold): ynew := g(xold,yold,zold): znew := h(xold,yold,zold):
mylist := mylist,[xnew,ynew,znew]:
xold := xnew: yold := ynew: zold := znew:
od:
> spacecurve([mylist],color=blue,axes=boxed);
Problem: Type the command lines listed above into Maple in the order in which they are listed. (Note:
You might find it useful to limit the range of N until you are sure your code has been typed correctly.) Then,
after experimenting with this code, discuss the effects that changing a, b, c and x0, y0, z0 have on this system.
Illustrate your discussion with Maple graphics.