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?