I am stumped on how to answer the following question:
Consider the Hénon map given by $$\textbf{H}(x, y) = (a-x^2+by, x)$$
Assume $0<b<1$. Classify the bifurcations that occur at $a = -\frac{1}{4}(1 - b)^2$ and $a = \frac{3}{4}(1 - b)^2$ by type and location of fixed and periodic points, if they exist.
I know how to find the fixed points, setting $\textbf{H}(x, y) = (a-x^2+by, x) = (x, y)$ and solving for $x$, getting $$x = \frac{1}{2}\left(b + 1 \pm \sqrt{(1 - b)^2 + 4a}\right)$$
Plugging in the given values of $a$ gives that the fixed point at $a = -\frac{1}{4}(1 - b)^2$ is at $x = \frac{b + 1}{2}$ and that the fixed points at $a = \frac{3}{4}(1 - b)^2$ are at $x = \frac{3 - b}{2}$ and $x = \frac{3b - 1}{2}$.
My questions now are how to find any periodic points and how to classify the bifurcations. Finding a periodic point, say, of period $n$, would involve solving $\textbf{H}^n{(x, y)} = (x, y)$, but I don't know how to find a closed form for $\textbf{H}^n (x, y)$. Is there any way to tell if there are no periodic points?
Any clarification would be appreciated.
