- 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?