FINITE DIFFERENCE EQUATIONS

1. A differential equation    dy/dt = f(y)   corresponds to a family of  difference equation

Y(n+1) - Y(n) =  h f(Y(n)).

If  h is small, the solutions should look something like the solutions to the differential equation,
but if  h is large, they may be very different.   Try converting the logistics equation  into a
finite difference equation with h = 1.     The  logistics equation has  athe form                dy/dt =  ry(1 - y).

2.  Find a solution to Y(n+1) =  a Y(n)  in terms of Y(0) with a = 1/2 and a = 2.  What happens to
the solutions as n goes to infinity?

3.   The logistics  difference equation has the form  Y(n+1) =  s Y(n)( 1 - Y(n)).  If is s is near one,
it is supposed to have the general behavior of the differential equation. Chaos occurs
when s > a special    value between 3 and 4.    What are the fixed points of this equation? See the
netmath  section on chaos for experiments.

4.  Suppose you are given the finite difference equation Y(n+1) = 3Y(n) (Y(n) - 1).
Let U(n) = Y(2n).   Find a finite difference equation for U(n).

5.  Find the fixed point for the equation you found in 4.   Why are the fixed points for  the Y
equation also fixed points for the U equation?