I've been investigating into situations where the N-R method iterates between two $x$ values endlessly. So far I have derived that the relationship between the two values of $x$ should be as follows: $$\frac{f(x_{even})}{f'(x_{even})} = -\frac{f(x_{odd})}{f'(x_{odd})}$$
where $x_0$ is the initial value of $x$ to be taken.
This can be derived as follows:
Let $a$ denote the even $x$ iterations and let $b$ denote the odd $x$ iterations.
All values of $a$ will be equal as will all values of $b$ because the method returns back to the same point an even amount of iterations later.
Hence, using the forumla, $$x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)},$$ we can gather that,
$$b=a-\frac{f(a)}{f'(a)}$$
and,
$$a=b-\frac{f(b)}{f'(b)}$$
Re-arranging these gives the result, $$\frac{f(a)}{f'(a)}=-\frac{f(b)}{f'(b)}$$
as required.
I'm curious as to what general equations/types of curves there would be in which this situation would occur (if any) where there are real roots to be found, as I'm having little luck finding out myself.
Newton's method is often understood as "riding the tangent line" to the root. Thus, the following image might represent two Newton steps from the green initial point:
From here, a simple continuity argument based on the following animation indicates that period 2 orbits often occur near an extreme that misses the $x$-axis: