Suppose that G: $\mathbb R$ $\rightarrow$ $\mathbb R$ such that G(x)=4x(1-x).
I need to find the period 2 orbit(s?) of this map and decide if it's a sink or a source. If I could find the fixed points under two iterations, this would be no problem for me, since I know how to determine if a given hyperbolic point is an attracting or repelling point. My problem is that the second iteration of this function will be a polynomial of order 4, and the rational root theorem shows that there are no rational roots, so I'm at a loss.
How can I go about finding these periodic points?
If $G(x) = 4x(1 - x)$, then
$$G(G(x)) = 4(4x(1 - x))(1 - 4x(1 - x)) = 16x(1 - x)(1 - 4x + 4x^2) = -64x^4 + 128x^3 - 80x^2 + 16x$$
Subtract $x$ because you want to solve $G(G(x)) = x$ which is the same as $G(G(x)) - x = 0$, and form the polynomial equation
$$-64x^4 + 128x^3 - 80x^2 + 15x = 0$$
Note you can divide by $x$ to get a cubic. Therefore we already have one solution, $x = 0$. Checking shows it is a fixed point. The cubic is
$$-64x^3 + 128x^2 - 80x + 15 = 0$$
The RRT gives possible numerators as 1, 3, 5, and 15, and possible denominators as $2^n$ for n from 0 to 6, of any sign. Hint: Take $4$ as a denominator.