Logistics Questions?

Question

The logistics map is the function f:[0,1] -> R, defined by f(x)=ax(1-x). Thus, x of n+1 = f(x of n) {Note, I say x of n+1 to represent x with the subscript n+1} and (x of n) = (x of 1, f(x of 1), f(f(x of 1)), ...). We write f^2(x) for f(f(x)), and so on.

Check that f has a global maximum of a/4 at x=1/2. Deduce that f maps [0,1] back into [0,1] if 0 <= x <= 4

From the above, we restrict ourselves to seeds, x of 1, from (0,1) and a epsilon (0,4). In fact the zone of a values in (0,1) is also not so interesting - the population is doomed to extinction.

Im having trouble with the deuce part. Thanks

Answer 1

Consider the shape of the logistics map.

It has a single hump, with a maximum value of a/4 at x = 1/2. At the same time, for x in [0,1] its value is always going to be greater than or equal to 0.

That means that for all x in [0,1] f(x) is going to be in [0, a/4].

If 0 <= a <= 4, then 0 <= a/4 <= 1, so
0 <= f(x) <= a/4 means 0 <= f(x) <= 1

That's it.